If the vector function r(t) is perpendicular to vector v for all values of t, then the derivative of r(t), denoted as r'(t), is also perpendicular to v for all t.
Furthermore, if r'(t) is perpendicular to v for all t, then r(t) lies in a plane that is perpendicular to v, and there exists a constant d such that the coordinates of r(t), denoted as ⟨x, y, z⟩, satisfy the equation ax + by + cz = d.
3.1. To show that r'(t) is perpendicular to v for all t when r(t) is perpendicular to v for all t, we can use the property that the derivative of a constant vector is the zero vector. Suppose r(t) is perpendicular to v for all t. Then, for any value of t, the dot product of r(t) and v is zero. Taking the derivative of both sides with respect to t, we get the derivative of the dot product of r(t) and v, which is the dot product of r'(t) and v. Since the dot product of r(t) and v is always zero, its derivative, r'(t) · v, must also be zero. Thus, r'(t) is perpendicular to v for all t.
3.2. Now, let's assume that r'(t) is perpendicular to v for all t. We need to show that there exists a constant d such that ax + by + cz = d, where ⟨x, y, z⟩ is a point on the vector function r(t). Taking the dot product of r'(t) and v, we have r'(t) · v = ⟨x', y', z'⟩ · ⟨a, b, c⟩ = ax' + by' + cz' = 0, where x', y', and z' are the derivatives of x, y, and z with respect to t, respectively. This equation represents a plane in 3D space. By integrating the equation, we can find that ax + by + cz = d, where d is a constant. Hence, r(t) lies in a plane perpendicular to v, and the equation ax + by + cz = d holds for all points ⟨x, y, z⟩ on r(t).
Learn more about vector function here:
https://brainly.com/question/33066980
#SPJ11
If a licensee is paid a salary of $600 per month plus half of the office's 6% fee on all sales, what MUST the licensee's monthly sales be in order for the licensee to receive a total monthly income of $2,220?
The licensee's monthly sales must be $54,000 for them to receive a total monthly income of $2,220.
To determine the monthly sales required for the licensee to receive a total monthly income of $2,220, we need to break down the components of the income.
Let's assume the total monthly sales amount to be x.
The licensee's income consists of two parts:
1. A salary of $600 per month.
2. Half of the office's 6% fee on all sales.
The office's fee on all sales can be calculated as (6/100) * x = 0.06x.
Therefore, the licensee's income from the office's fee on all sales is (1/2) * 0.06x = 0.03x.
Adding the salary and the income from the office's fee, the licensee's total monthly income is given by:
$600 + 0.03x = $2,220.
To find the value of x, we need to solve this equation:
0.03x = $2,220 - $600,
0.03x = $1,620.
Dividing both sides by 0.03, we get:
x = $1,620 / 0.03,
x = $54,000.
Therefore, the licensee's monthly sales must be $54,000 for them to receive a total monthly income of $2,220.
Learn more about monthly income here:
https://brainly.com/question/29207678
#SPJ11
The licensee's monthly sales must reach $54,000 for the licensee to receive a total monthly monthly income of $2,220. This is determined by subtracting the licensee's base salary from the total desired income and calculating the sales that would result in the remaining amount as half the 6% sales fee.
Explanation:To determine the licensee's monthly sales for the licensee to receive a total monthly income of $2,220, we must first deduce the part of the income that comes from the licenses' share of the 6% fee on sales.
To do this, we subtract the licensee's base salary, which is $600, from the total desired income of $2,220. This gives us $2,220 - $600 = $1,620.
Since this $1,620 represents half of the 6% fee on sales, it means the full 6% of sales is $1,620 * 2 = $3,240. From this, we can calculate the actual sales since we know that 6% of the sales is equal to $3,240.
Therefore, to find the total sales, we divide $3,240 by 0.06 (which is 6% in decimal form). That gives us $3,240 / 0.06 = $54,000.
So, the licensee's monthly sales must be $54,000 for the licensee to receive a total monthly income of $2,220.
Learn more about Fractional Sales Calculation here:https://brainly.com/question/28291629
#SPJ2
A manufacturing company in city A wishes to truck its product to 4 different cities, B,C,D, and E. If the cities are all interconnected by roads, how many different route plans can be constructed so that a single truck, starting from A, will visit each city exactly once, then return home?
A manufacturing company in city A needs to find 120 route plans to truck its product to four interconnected cities. Using permutations, the total number of ways to arrange the cities is 5!, with the order of the cities not crucial.
Given that a manufacturing company in city A wishes to truck its product to 4 different cities, B, C, D, and E. If the cities are all interconnected by roads, then we need to find the number of different route plans that can be constructed so that a single truck, starting from A, will visit each city exactly once, then return home.
To find the required solution, we will use the concept of permutations. We know that if we have n items, then the number of ways we can arrange them in a row is n! (n factorial).
Hence, the total number of ways to arrange the 5 cities is 5!. However, the order in which we arrange these 5 cities does not matter. There are 5 choices for the first city, 4 choices for the second city, 3 choices for the third city, 2 choices for the fourth city, and only one choice left for the last city. So, the total number of different route plans that can be constructed is given as follows:
5! / (1! × 1! × 1! × 1! × 1!)
= 5 × 4 × 3 × 2 × 1 / 1 × 1 × 1 × 1 × 1
= 120
Therefore, there are 120 different route plans that can be constructed so that a single truck, starting from A, will visit each city exactly once, then return home.
To know more about permutations Visit
https://brainly.com/question/32683496
#SPJ11
2. Given that the linear system Ax=b has a particular solution p. Show that for every solution y of Ax=b, there is a solution v of the homogeneous linear system Ax=0 such that y=p+v. Hint: Consider y−p.
This proves that for every solution y of Ax = b, there is a solution v of the homogeneous linear system Ax = 0 such that y = p + v.
Given that the linear system Ax = b has a particular solution p.
We are supposed to show that for every solution y of Ax = b, there is a solution v of the homogeneous linear system Ax = 0 such that y = p + v.
Hint: Consider y - p.
To prove this, we can consider the difference between the two solutions y and p and take that as our solution v of Ax = 0.
Since p is a solution to Ax = b,
it follows that Ap = b.
Since y is also a solution to Ax = b,
it follows that Ay = b.
We can subtract the two equations to get:
Ay - Ap = 0 which gives us:
A(y - p) = 0
So, the solution to Ax = 0 is y - p,
which means that there exists some vector v such that Av = 0 and y - p = v.
Therefore, we have y = p + v where v is a solution of Ax = 0.
Hence, this proves that for every solution y of Ax = b, there is a solution v of the homogeneous linear system Ax = 0 such that y = p + v.
To know more about linear system visit:
https://brainly.com/question/26544018
#SPJ11
mm 45 degree elbow and is diverted upwards into a connecting pipe. The diameter of the outlet is 85mm. If the pressure just before the elbow is 95 kPa, determine the magnitude and direction of the force necessary to keep the elbow in place. Assume frictional losses in the elbow are negligible.
To determine the magnitude and direction of the force necessary to keep the 45-degree elbow in place, we need to consider the pressure difference across the elbow and the momentum change of the fluid.
Neglecting frictional losses, the force can be calculated using the principles of fluid mechanics. The force required to keep the elbow in place can be determined by analyzing the pressure difference across the elbow and the momentum change of the fluid.
First, let's consider the pressure difference. Since frictional losses are negligible, the pressure remains constant throughout the fluid flow. Given that the pressure just before the elbow is 95 kPa, the pressure at the outlet will also be 95 kPa.
Next, we need to analyze the momentum change of the fluid. As the fluid flows through the 45-degree elbow, it changes direction. This change in momentum creates a force that acts on the elbow. According to Newton's second law, force is equal to the rate of change of momentum. In this case, the force required to keep the elbow in place is equal to the change in momentum of the fluid.
To calculate the magnitude of the force, we can use the principle of conservation of momentum. Assuming the fluid is incompressible and the flow is steady, the momentum before the elbow is equal to the momentum after the elbow. By considering the fluid velocity and density, we can calculate the momentum change and thus determine the magnitude of the force.
To determine the direction of the force, we need to consider the change in fluid momentum. As the fluid flows through the 45-degree elbow and is diverted upwards, its momentum changes direction. Therefore, the force required to keep the elbow in place will act in the opposite direction of the momentum change, which is downward.
In summary, the force necessary to keep the 45-degree elbow in place can be calculated by considering the pressure difference across the elbow and the momentum change of the fluid. The magnitude of the force can be determined by analyzing the momentum change of the fluid, assuming steady flow and neglecting frictional losses. The direction of the force will be opposite to the momentum change, which in this case is downward.
To learn more about direction refer:
https://brainly.com/question/29248951
#SPJ11
6. You are given that \( u \) is an angle in the third quadrant and that sin \( u=-\frac{5}{13} \). (a) Draw a right-ingled triangle for \( u \). \( (2) \) This question is contimued on page 3 2 (b) C
Let's construct the right triangle and fill in the lengths: triangle image First of all, we have the information that the sine of the angle is -5/13, and we are looking at an angle in the third quadrant.
Since sine is negative in the third quadrant, we know that the angle has a reference angle (the angle between the terminal side and the x-axis) in the first quadrant that gives the same sine value. We can use the Pythagorean theorem to find the length of the hypotenuse, since we know that:
Since the adjacent side has length 0, we don't even need to draw it in the triangle. We can just draw a vertical line for the opposite side, and a horizontal line for the hypotenuse. The length of the hypotenuse is:[tex]$$c = -\frac{13}{5} \cdot[/tex]opposite [tex]= -\frac{13}{5} \cdot 5[/tex]
=[tex]-13$$[/tex] So the right triangle looks like.
To know more about triangle visit:
https://brainly.com/question/2773823
#SPJ11
Use DeMoivre's Theorem to find (−1+√3i)^12
Write the answer in the form of a + bi
DeMoivre's Theorem is a useful mathematical formula that can help to find the powers of complex numbers. It uses trigonometric functions to determine the angle and magnitude of the complex number.
This theorem states that for any complex number `z = a + bi`, `z^n = r^n (cos(nθ) + i sin(nθ))`.Here, `r` is the modulus or magnitude of `z` and `θ` is the argument or angle of `z`.
Let's apply DeMoivre's Theorem to find `(−1+√3i)^12`.SolutionFirst, we need to find the modulus and argument of the given complex number.`z = -1 + √3i`Magnitude or modulus `r = |z| = sqrt((-1)^2 + (√3)^2) = 2`Argument or angle `θ = tan^-1(√3/(-1)) = -π/3`Now, let's find the power of `z^12` using DeMoivre's Theorem.`z^12 = r^12 (cos(12θ) + i sin(12θ))``z^12 = 2^12 (cos(-4π) + i sin(-4π))`Since cosine and sine are periodic functions, their values repeat after each full cycle of 2π radians or 360°.
Therefore, we can simplify the expression by subtracting multiple of 2π from the argument to make it lie in the range `-π < θ ≤ π` (or `-180° < θ ≤ 180°`).`z^12 = 2^12 (cos(2π/3) + i sin(2π/3))``z^12 = 4096 (-1/2 + i √3/2)`Now, we can express the answer in the form of `a + bi`.Multiplying `4096` with `-1/2` and `√3/2` gives:`z^12 = -2048 + 2048√3i`Hence, `(−1+√3i)^12 = -2048 + 2048√3i`.Conclusion:Thus, using DeMoivre's Theorem, we have found that `(−1+√3i)^12 = -2048 + 2048√3i`
To know more about DeMoivre's Theorem visit
https://brainly.com/question/28035659
#SPJ11
Show that the second-order wave equation δu²/δt² = c² δ²u/δx² is a hyperbolic equation
The hyperbolic equations can be represented as the second-order partial differential equations, which have two different characteristics in nature. These equations can be obtained by finding the solution for the Laplace equation with variable coefficients, which are used to describe the behavior of a certain physical system such as wave propagation, fluid flow, or heat transfer.
The second-order wave equation δu²/δt² = c² δ²u/δx² is a hyperbolic equation since it can be obtained by finding the solution of the Laplace equation with variable coefficients. The wave equation is a second-order partial differential equation that describes the behavior of waves. It has two different characteristics in nature, which are represented by two independent solutions.The first solution is a wave traveling to the right, while the second solution is a wave traveling to the left.
The equation is hyperbolic since the characteristics of the equation are hyperbolic curves that intersect at a point. This intersection point is known as the wavefront, which is the location where the wave is at its maximum amplitude.The wave equation has many applications in physics, engineering, and mathematics.
It is used to describe the behavior of electromagnetic waves, acoustic waves, seismic waves, and many other types of waves. The equation is also used in the study of fluid dynamics, heat transfer, and other fields of science and engineering. Overall, the second-order wave equation is a hyperbolic equation due to its characteristics, which are hyperbolic curves intersecting at a point.
To know more about equations visit:
https://brainly.com/question/29538993
#SPJ11
Convert these values to scientific notation.
Part 1 (1 point)
log x = 11.51 ; x
= Part 2 (1 point)
log x = -8.95 ; x
=
The coefficient is a value greater than or equal to 1 but less than 10, and the power indicates the number of decimal places the decimal point should be moved
Part 1:
The value of x can be calculated using the logarithmic function. Given log x = 11.51, we can rewrite it in exponential form as x = 10^11.51. In scientific notation, this can be expressed as x = 3.548 × 10^11.
Part 2:
Similarly, for log x = -8.95, we can rewrite it in exponential form as x = 10^(-8.95). In scientific notation, this can be expressed as x = 3.125 × 10^(-9).
Learn more about values here : brainly.com/question/30145972
#SPJ11
Find the effective yield of an investment that earns 2.75% compounded daily. round to the nearest hundredth of a percent
The effective yield of the investment is approximately 2.81% when rounded to the nearest hundredth of a percent.
To calculate the effective yield of an investment that earns 2.75% compounded daily, we can use the following formula:
Effective Yield = (1 + (Nominal Interest Rate / Number of Compounding Periods))^Number of Compounding Periods - 1
In this case, the nominal interest rate is 2.75% and it is compounded daily, which means there are 365 compounding periods in a year (assuming non-leap year).
Plugging in the values into the formula, we get:
Effective Yield = (1 + (0.0275 / 365))^365 - 1
Calculating this expression, we find:
Effective Yield ≈ 0.028085159 - 1 ≈ 0.0281
Therefore, the effective yield of the investment is approximately 2.81% when rounded to the nearest hundredth of a percent.
Learn more about Interest here:
https://brainly.com/question/26457073
#SPJ11
A baseball is hit so that its height in feet t seconds after it is hit can be represented by the following. (Include units in your answers. More information.) h(t)=−16t2+67t+2.5 (a) What is the height of the ball when it is hit? The ball is at a height of when it is hit. (b) When does the ball reach a height of 40 feet? (Round your answers to the nearest hundredth of a second.) The ball reached a height of 40 feet about (smaller value) and again (larger value) after being hit. (c) What is the ball's maximum height? (Round your answer to the nearest hundredth of a foot.) The ball reached a maximum height of (d) If it is not caught, when does the ball hit the ground? (Round your answer to the nearest hundredth of a second.) The ball hits the ground at
The initial height of the baseball can be calculated by substituting t = 0 into the given equation:h(0) = -16(0)^2 + 67(0) + 2.5= 2.5 Therefore, the ball is at a height of 2.5 feet when it is hit.
To find when the ball reaches a height of 40 feet, we need to solve the following equation for t:-16t^2 + 67t + 2.5 = 40Using the quadratic formula, we can get the two possible values of t as follows:t ≈ 1.09 and t ≈ 4.74Therefore, the ball reached a height of 40 feet about 1.09 seconds and again 4.74 seconds after being hit.
The maximum height of the baseball occurs at the vertex of the parabolic path, which is given by the formula:t = -b / 2a = -67 / 2(-16) = 2.09Using this value of t in the equation, we can get the maximum height as follows:h(2.09) = -16(2.09)^2 + 67(2.09) + 2.5 ≈ 82.14Therefore, the ball reached a maximum height of 82.14 feet.d. To find when the ball hits the ground, we need to find the value of t when h(t) = 0. Therefore, we need to solve the following equation for t:-16t^2 + 67t + 2.5 = 0Using the quadratic formula, we can get the two possible values of t as follows:t ≈ 0.16 and t ≈ 4.18Therefore, the ball hits the ground at about 0.16 seconds and again 4.18 seconds after being hit.
To know more about calculated visit :
https://brainly.com/question/30035551
#SPJ11
Find all EXACT solutions of the equation given below in the interval \( [0, \pi) \). \[ \sin (3 x)=-\frac{\sqrt{3}}{2} \] If there is more than one answer, enter them in a list separated by commas. En
The required exact solutions of this equation are [tex]$$\boxed{\frac{4\pi}{9}, \frac{5\pi}{9}, \frac{16\pi}{9}, \frac{17\pi}{9}}$$[/tex]
The given equation is
[tex]$\sin(3x)=-\frac{\sqrt{3}}{2}$.[/tex]
The first step to solving this equation is to solve for [tex]$3x$[/tex].
We know that
[tex]$\sin(60^o) = \frac{\sqrt{3}}{2}$,[/tex] so we need to find the angle whose sine is
[tex]$-\frac{\sqrt{3}}{2}$[/tex] (since $\sin$ is negative in the third and fourth quadrants).
This angle will be [tex]$240°$[/tex] since [tex]$\sin(240^o) = -\frac{\sqrt{3}}{2}$[/tex].
The reference angle for $240°$ is $60°$, which is the same as the reference angle for [tex]$\frac{\sqrt{3}}{2}$[/tex].
Since the sine function is negative in the third and fourth quadrants, we must add $180°$ to each solution to get the angles in the interval $[0, \pi)$.
Hence, we have:
[tex]$$\begin{aligned} 3x&=\frac{4\pi}{3}+360^on\\ 3x&=\frac{5\pi}{3}+360^om \end{aligned}$$[/tex]
where $n, m$ are any integer.
Find exact solutions by solving for [tex]$x$[/tex] in each equation.
We get: [tex]$$\begin{aligned} x&=\frac{4\pi}{9}+120^on\\ x&=\frac{5\pi}{9}+120^om \end{aligned}$$[/tex]
where $n, m$ are any integer.
Since the interval is[tex]$[0, \pi)$[/tex], we only need to consider the values of [tex]$[0, \pi)$[/tex] and [tex]$m$[/tex] that make [tex]$x$[/tex] in this interval.
To know more about quadrants , visit:
https://brainly.in/question/28784290
#SPJ11
The exact solution is [tex]$x=\frac{2\pi}{9}$[/tex] (in radians). The required solution is: [tex]$\frac{2\pi}{9}$[/tex].
The given equation is:
[tex]$ \sin (3 x)=-\frac{\sqrt{3}}{2} $[/tex]
The interval is [tex]$[0, \pi)$[/tex]
To solve for x, use inverse sine function on both sides:
[tex]\[\begin{aligned}\sin (3 x)&=-\frac{\sqrt{3}}{2} \\ \sin^{-1} \sin (3 x)&=\sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) \\ 3 x &= -\frac{\pi}{3} + k \pi \quad \text{or} \quad 3 x = \frac{2\pi}{3} + k \pi, \quad \text{where} \quad k\in \mathbb{Z}\end{aligned}\][/tex]
To get the values of x in the interval [tex]$[0, \pi)$[/tex]:
For
[tex]$3x = -\frac{\pi}{3}$[/tex]
we have [tex]$x = -\frac{\pi}{9}$[/tex],
which is outside the given interval.
For [tex]$3 x = \frac{2\pi}{3}$[/tex],
we have [tex]$x = \frac{2\pi}{9}$[/tex],
which is within the given interval.
So, the exact solution is [tex]$x=\frac{2\pi}{9}$[/tex] (in radians).
Therefore, the required solution is: [tex]$\frac{2\pi}{9}$[/tex].
To know more about inverse sine function, visit:
https://brainly.com/question/24160092
#SPJ11
4. Let f : A → B.
(a) Decide if the following statement is true or false, and prove your answer: for all subsets S and T of A, f(S \ T) ⊆ f(S) \ f(T). If the statement is false, decide if the assumption that f is one-to-one, or that f is onto, will make the statement true, and prove your answer.
(b) Repeat part (a) for the reverse containment.
(a) The statement f(S \ T) ⊆ f(S) \ f(T) is false and here is the proof:
Let A = {1, 2, 3}, B = {4, 5}, and f = {(1, 4), (2, 4), (3, 5)}.Then take S = {1, 2}, T = {2, 3}, so S \ T = {1}, then f(S \ T) = f({1}) = {4}.
Moreover, we have f(S) = f({1, 2}) = {4} and f(T) = f({2, 3}) = {4, 5},thus f(S) \ f(T) = { } ≠ f(S \ T), which implies that the statement is false.
Then to show that the assumption that f is one-to-one, or that f is onto, will make the statement true, we can consider the following two cases. Case 1: If f is one-to-one, the statement will be true.We will prove this statement by showing that f(S \ T) ⊆ f(S) \ f(T) and f(S) \ f(T) ⊆ f(S \ T).
For f(S \ T) ⊆ f(S) \ f(T), take any x ∈ f(S \ T), then there exists y ∈ S \ T such that f(y) = x. Since y ∈ S, it follows that x ∈ f(S).
Suppose that x ∈ f(T), then there exists z ∈ T such that f(z) = x.
But since y ∉ T, we get y ∈ S and y ∉ T,
which implies that z ∉ S.
Thus, we have f(y) = x ∈ f(S) \ f(T).
Therefore, f(S \ T) ⊆ f(S) \ f(T).For f(S) \ f(T) ⊆ f(S \ T),
take any x ∈ f(S) \ f(T), then there exists y ∈ S such that f(y) = x, and y ∉ T. Thus, y ∈ S \ T, and it follows that x = f(y) ∈ f(S \ T).
Therefore, f(S) \ f(T) ⊆ f(S \ T).
Thus, we have shown that f(S \ T) ⊆ f(S) \ f(T) and f(S) \ f(T) ⊆ f(S \ T), which implies that f(S \ T) = f(S) \ f(T) for all subsets S and T of A,
when f is one-to-one.
Case 2: If f is onto, the statement will be true.
We will prove this statement by showing that f(S \ T) ⊆ f(S) \ f(T) and f(S) \ f(T) ⊆ f(S \ T).For f(S \ T) ⊆ f(S) \ f(T),
take any x ∈ f(S \ T), then there exists y ∈ S \ T such that f(y) = x.
Suppose that x ∈ f(T), then there exists z ∈ T such that f(z) = x.
But since y ∉ T, it follows that z ∈ S, which implies that x = f(z) ∈ f(S). Therefore, x ∈ f(S) \ f(T).For f(S) \ f(T) ⊆ f(S \ T), take any x ∈ f(S) \ f(T),
then there exists y ∈ S such that f(y) = x, and y ∉ T. Since f is onto, there exists z ∈ A such that f(z) = y.
Thus, z ∈ S \ T, and it follows that f(z) = x ∈ f(S \ T).
Therefore, x ∈ f(S) \ f(T).Thus, we have shown that f(S \ T) ⊆ f(S) \ f(T) and f(S) \ f(T) ⊆ f(S \ T), which implies that f(S \ T) = f(S) \ f(T) for all subsets S and T of A, when f is onto.
The statement f(S \ T) ⊆ f(S) \ f(T) is false. The assumption that f is one-to-one or f is onto makes the statement true.(b) Repeat part (a) for the reverse containment.Since the conclusion of part (a) is that f(S \ T) = f(S) \ f(T) for all subsets S and T of A, when f is one-to-one or f is onto, then the reverse containment f(S) \ f(T) ⊆ f(S \ T) will also hold, and the proof will be the same.
Learn more about one-to-one here:
brainly.com/question/31777644
#SPJ11
heights of adults. researchers studying anthropometry collected body girth measurements and skele- tal diameter measurements, as well as age, weight, height and gender, for 507 physically active individuals. the histogram below shows the sample distribution of heights in centimeters.8 100 80 60 40 20 0 min 147.2 q1 163.8 median 170.3 mean 171.1 sd 9.4 q3 177.8 max 198.1 150 160 170 180 height 190 200 (a) what is the point estimate for the average height of active individuals? what about the median? (b) what is the point estimate for the standard deviation of the heights of active individuals? what about the iqr? (c) is a person who is 1m 80cm (180 cm) tall considered unusually tall? and is a person who is 1m 55cm (155cm) considered unusually short? explain your reasoning. (d) the researchers take another random sample of physically active individuals. would you expect the mean and the standard deviation of this new sample to be the ones given above? explain your reasoning. (e) the sample means obtained are point estimates for the mean height of all active individuals, if the sample of individuals is equivalent to a simple random sample. what measure do we use to quantify the variability of such an estimate? compute this quantity using the data from the original sample under the condition that the data are a simple random sample.
The standard error for the mean height estimate is approximately 0.416 centimeters.
(a) The point estimate for the average height of active individuals is 171.1 centimeters, which is equal to the mean height of the sample. The median height, on the other hand, is 170.3 centimeters, which represents the midpoint of the sorted sample.
(b) The point estimate for the standard deviation of the heights of active individuals is 9.4 centimeters, which is equal to the standard deviation of the sample. The interquartile range (IQR) can be determined from the values given in the histogram. It is the difference between the third quartile (Q3) and the first quartile (Q1), which yields an IQR of 177.8 - 163.8 = 14 centimeters.
(c) To determine if a person's height is considered unusually tall or short, we can examine their position relative to the measures of central tendency and spread. A person who is 180 cm tall falls within one standard deviation of the mean height (171.1 ± 9.4 cm) and is not considered unusually tall. Similarly, a person who is 155 cm tall falls within one standard deviation below the mean and is not considered unusually short.
(d) When another random sample of physically active individuals is taken, we would expect the mean and standard deviation of this new sample to be similar to the ones given above. This is because the sample statistics (mean and standard deviation) provide estimates of the population parameters (mean and standard deviation), and with a random sample, the estimates tend to converge to the true population values as the sample size increases.
(e) The measure we use to quantify the variability of the estimate (mean height) based on a simple random sample is the standard error. The standard error can be calculated as the standard deviation of the sample divided by the square root of the sample size. Using the data from the original sample (sample size = 507, standard deviation = 9.4), we can compute the standard error as:
Standard Error = 9.4 / sqrt(507) ≈ 0.416
know more about standard error here:
https://brainly.com/question/32854773
#SPJ11
A boat heads
38°,
propelled by a force of
850
lb. A wind from
308°
exerts a force of
175
lb on the boat. How large is the resultant force
F,
and in what direction is the boat moving?
1.The magnitude of the resultant force F is
(round nearest integer as needed)
2.
The direction the boat is moving is
1. The magnitude of the resultant force \(F\) is 890 lb (rounded to the nearest integer).
2. The direction the boat is moving is 2° north of east.
To find the resultant force, we can use vector addition. The force exerted by the boat can be represented as a vector of magnitude 850 lb in the direction 38° east of north. The force exerted by the wind can be represented as a vector of magnitude 175 lb in the direction 52° west of north (308° clockwise from north).
To find the resultant force, we can add these two vectors using vector addition. The magnitude of the resultant force can be found using the law of cosines:
[tex]\[F^2 = (850)^2 + (175)^2 - 2 \cdot 850 \cdot 175 \cdot \cos(90° - (52° - 38°))\][/tex]
Simplifying this expression, we find \(F \approx 890\) lb.
To determine the direction the boat is moving, we can use the law of sines:
[tex]\[\sin(\text{{direction of resultant force}}) = \frac{175 \cdot \sin(90° - 52°)}{890}\][/tex]
Solving for the direction, we find the boat is moving 2° north of east.
Therefore, the magnitude of the resultant force \(F\) is 890 lb and the boat is moving 2° north of east.
learn more about resultant force here
https://brainly.com/question/22260425
#SPJ11
the
number of 3 digit numbers less than 500 that can be created if the
last digit is either 4 or 5 is?
To find the number of 3 digit numbers less than 500 that can be created if the last digit is either 4 or 5 we can use the following steps:
Step 1: Numbers less than 500 are 100, 101, 102, 103, ... 499
Step 2: The last digit of the number is either 4 or 5 i.e. {4, 5}. Therefore, we have 2 options for the last digit.
Step 3: For the first two digits, we can use any of the digits from 0 to 9. Since the number of options is 10 for both digits, the total number of ways we can choose the first two digits is 10 × 10 = 100.
Step 4: Hence, the total number of 3 digit numbers less than 500 that can be created if the last digit is either 4 or 5 is 2 × 100 = 200.
Therefore, the number of 3 digit numbers less than 500 that can be created if the last digit is either 4 or 5 is 200.
To know more about digit visit :-
https://brainly.com/question/26856218
#SPJ11
The tallest radio tower in the United States is in the Oro Valley near Tucson, Arizona. A cable from its top attached to the ground 260 feet from its base is 700 feet long. How tall is the radio tower
The height of the radio tower is 600 feet.
we can use the Pythagorean theorem. According to the Pythagorean theorem, In a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Let the height of the radio tower be x feet. The length of the cable is 700 feet. The length of the horizontal side is 260 feet.
Therefore, according to the Pythagorean theorem,
[tex]\[\left( {x} \right)^2= {\left( {700} \right)^2} - {\left( {260} \right)^2}\][/tex]
After substituting the given values, we get
[tex]\[\left( {x} \right)^2 = \left( {490000} \right) - \left( {67600} \right)\][/tex]
[tex]\[\left( {x} \right)^2 = \left( {422400} \right)\][/tex]
Thus, [tex]\[x = \sqrt {422400}\]\[/tex]
[tex]\[x= 600\][/tex]
Hence, the height of the radio tower is 600 feet.
to know more Pythagorean theorem
https://brainly.com/question/14930619
#SPJ11
doubling time of fles is 4 how s What factor does pop. uncrease in 28 horns ∀ what factor increase in 2 weeks? 4
8
12
16
20
24
28
2x
4x
8x
16x
32x
64x
128x
The population will increase by a factor of 16 in 28 hours, and by a factor of 128 in 2 weeks.
If the doubling time of a population is 4 hours, it means that the population doubles every 4 hours. Therefore, in 28 hours, the population would double 7 times (28 divided by 4), resulting in an increase of 2^7, which is 128. So the population would increase by a factor of 128 in 28 hours.
Similarly, to determine the population increase in 2 weeks, we need to convert the time to hours. There are 24 hours in a day, so 2 weeks (14 days) would be equal to 14 multiplied by 24, which is 336 hours. Since the doubling time is 4 hours, the population would double 336 divided by 4 times, resulting in an increase of 2^(336/4), which is 2^84. Simplifying, this is equal to 2^(4*21), which is 2^84. Therefore, the population would increase by a factor of 128 in 2 weeks.
In summary, the population would increase by a factor of 16 in 28 hours and by a factor of 128 in 2 weeks.
Learn more about population here:
https://brainly.com/question/31598322
#SPJ11
The pH scale for acidity is defined by pH = -log[H+] where [H+] is the concentration of hydrogen ions measured in moles per liter (M). a) A sample of Pepsi is found to have a hydrogen concentration of 0.00126 M. What is the pH? pH= b) The pH of a sample of rhubarb is 3.4. What is the hydrogen concentration?
(a) The pH of the Pepsi sample is 2.9.
(b) The hydrogen concentration of the rhubarb sample is 0.000398107 M.
(a) To calculate the pH of the sample of Pepsi with a hydrogen ion concentration of 0.00126 M, we can use the formula:
pH = -log[H+]
Substituting the provided concentration:
pH = -log(0.00126)
Using logarithmic properties, we can calculate:
pH = -log(1.26 x 10^(-3))
Taking the logarithm:
pH = -(-2.9)
pH = 2.9
Therefore, the pH of the Pepsi sample with hydrogen concentration of 0.00126 M is 2.9.
(b) To calculate the hydrogen concentration of the sample of rhubarb with a pH of 3.4, we can rearrange the equation:
pH = -log[H+]
To solve for [H+], we take the antilog (inverse logarithm) of both sides:
[H+] = 10^(-pH)
Substituting the provided pH:
[H+] = 10^(-3.4)
[H+] = 0.000398107
Therefore, the hydrogen concentration of the rhubarb sample with pH of a sample of rhubarb is 3.4 is 0.000398107 M.
To know more about pH refer here:
https://brainly.com/question/2288405#
#SPJ11
Problem 2 Your ANS: Vectors The angles shown measure from the +x-axis to each vector. At what angle does the resultant make with the +x-axis, in degrees measured counterclockwise? 191 26 10 361 375
The angle that the resultant vector makes with the +x-axis is 603° measured counterclockwise.
How to find the angle that the resultant vectorTo find the angle that the resultant vector makes with the +x-axis, we need to add up the angles of the given vectors and find the equivalent angle in the range of 0 to 360 degrees.
Let's calculate the sum of the given angles:
191° + 26° + 10° + 361° + 375° = 963°
Since 963° is greater than 360°, we can find the equivalent angle by subtracting 360°:
963° - 360° = 603°
Therefore, the angle that the resultant vector makes with the +x-axis is 603° measured counterclockwise.
Learn more about angle at https://brainly.com/question/25716982
#SPJ4
Let X={1,3,5} and Y={s,t,u,v}. Define f:X→Y by the following arrow diagram. a. Write the domain of f and the co-domain of f. b. Find f(1),f(3), and f(5). c. What is the range of f ? 17. Define vertex set V, edge set E, order, size and degree sequence.
The domain of f is X and the co-domain of f is Y And f(1) = s, f(3) = t, f(5) = u. The range of f is {s, t, u}.
a. The domain of function f is X, which consists of the elements {1, 3, 5}. The co-domain of f is Y, which consists of the elements {s, t, u, v}.
b. Evaluating f(x) for each element in the domain, we have:
f(1) = s
f(3) = t
f(5) = u
c. The range of f represents the set of all possible output values. From the given information, we can see that f(1) = s, f(3) = t, and f(5) = u. Therefore, the range of f is the set {s, t, u}.
In graph theory, a graph consists of a vertex set V and an edge set E. The order of a graph is the number of vertices in the vertex set V. The size of a graph is the number of edges in the edge set E. The degree sequence of a graph represents the degrees of its vertices listed in non-increasing order.
To learn more about “graph” refer to the https://brainly.com/question/19040584
#SPJ11
Solve the triangle. \[ a=7.103 \text { in } c=6.127 \text { in } B=79.77^{\circ} \] What is the length of side \( b \) ? in (Round to the nearest thousandth as needed.) What is the measure of angle \(
To solve the triangle, we can use the Law of Sines, which states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant for all sides and angles in the triangle.
Let's label the triangle with sides \(a\), \(b\), and \(c\), and angles \(A\), \(B\), and \(C\), respectively.
Given:
[tex]\(a = 7.103\) in\(c = 6.127\) in\(B = 79.77^\circ\)[/tex]
We need to find the length of side \(b\) and the measure of angle \(A\).
Using the Law of Sines, we have:
[tex]\(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)[/tex]
Let's solve for side \(b\) first:
[tex]\(\frac{a}{\sin A} = \frac{b}{\sin B}\)[/tex]
Rearranging the equation, we get:
[tex]\(b = \frac{a \cdot \sin B}{\sin A}\)[/tex]
Plugging in the given values, we have:
[tex]\(b = \frac{7.103 \cdot \sin(79.77^\circ)}{\sin A}\)[/tex]
[tex]To find angle \(A\), we can use the fact that the sum of the angles in a triangle is \(180^\circ\):\(A + B + C = 180^\circ\)Substituting the given values, we have:\(A + 79.77^\circ + C = 180^\circ\)\(A + C = 180^\circ - 79.77^\circ\)\(A + C = 100.23^\circ\)[/tex]
[tex]Now, we can use the Law of Sines again to find angle \(A\):\(\frac{a}{\sin A} = \frac{c}{\sin C}\)Rearranging the equation, we get:\(\sin A = \frac{a \cdot \sin C}{c}\)Plugging in the given values, we have:\(\sin A = \frac{7.103 \cdot \sin(100.23^\circ)}{6.127}\)Now we can solve for angle \(A\) using the arcsine function:\(A = \arcsin\left(\frac{7.103 \cdot \sin(100.23^\circ)}{6.127}\right)\)\\[/tex]
Finally, we can calculate the value of side \(b\) by substituting the calculated values of \(A\) and \(B\) into the earlier equation:
[tex]\(b = \frac{7.103 \cdot \sin(79.77^\circ)}{\sin A}\)[/tex]
Round the values to the nearest thousandth as needed.
Please note that the exact values of \(A\) and \(b\) can be obtained using a calculator or software capable of performing trigonometric calculations.
To know more about cost click-
http://brainly.com/question/25109150
#SPJ11
pls help if you can asap!!
The correct option is the first one, the measure of angle B is 78°.
How to find the measure of angle B?On the diagram we can see an equilateral triangle, so the two lateral sides have the same length, so the two lateral angles have the same measure, that means that:
A = C
51° = C
Now remember that the sum of the interior angles of any trianglu must be 180°, then we can write:
A + B + C = 180°
51° + B + 51° = 180°
B = 180° - 102°
B = 78°
The corret option is the first one.
Learn more about angles at:
https://brainly.com/question/25716982
#SPJ1
a. If the function f:R→R is continuous, then f(R)=R. b. For any function f:[0,1]→R, its image f([0,1]) is an interval. c. For any continuous function f:D→R, its image f(D) is an interval. d. For a continuous strictly increasing function f:[0,1]→R, its image is the interval [f(0),f(1)].
a. False.The range of a continuous function can be a proper subset of R. b. True c. False d. True.
a. False. The statement is not true in general. While it is true that if a function f:R→R is continuous, then its range is a connected subset of R, it does not necessarily imply that the range is equal to the entire set of real numbers R. The range of a continuous function can be a proper subset of R, such as an interval, a single point, or even an empty set. b. True. The statement is true. For any function f:[0,1]→R, the image f([0,1]) is indeed an interval. This is a consequence of the Intermediate Value Theorem, which states that if a continuous function takes on two distinct values within an interval, then it must take on every value in between. Since [0,1] is a connected interval, the image of f([0,1]) must also be a connected interval.
c. False. The statement is not true in general. While it is true that continuous functions map connected sets to connected sets, it does not imply that the image of a continuous function on any domain D will always be an interval. The image can still be a proper subset of R, such as an interval, a single point, or even an empty set.
d. True. The statement is true. For a continuous strictly increasing function f:[0,1]→R, its image is indeed the interval [f(0),f(1)]. Since f is strictly increasing, any value between f(0) and f(1) will be attained by the function on [0,1]. Moreover, f(0) and f(1) themselves are included in the image since f is defined at both endpoints. Therefore, the image of f is the closed interval [f(0),f(1)].
To learn more about continuous function click here:
brainly.com/question/28228313
#SPJ11
How many solutions are there to the equation x₁ + x₂ + x3 + x₁ + x5 = 79 where the x, are nonnegative integers with ₁ ≥ 2, x3 ≥ 4, and 4 ≤ 7?
There are 3240 solutions for the equation x₁ + x₂ + x3 + x₁ + x5 = 79.
Given, x₁ + x₂ + x3 + x₁ + x5 = 79,
where the x are non-negative integers with ₁ ≥ 2, x3 ≥ 4, and 4 ≤ 7.
Therefore, x₂ = 0, x₄ = 0, and x₁, x₃, x₅ are the only variables.
Now, the equation is: x₁ + x₃ + x₅ = 79.
Using the method of stars and bars, the number of solutions is
(79+3-1) C (3-1) = 81 C 2 = (81 * 80) / 2 = 3240.
There are 3240 solutions.
Learn more about stars and bars visit:
brainly.com/question/31770493
#SPJ11
What is the surface area of the cuboid below?
Remember to give the correct units.
9m
12 m
✓ Scroll down
4 m
Not drawn accurately
Answer:
364 meters squared
Step-by-step explanation:
2(9*12+4*12+9*4) = 2(108+48+36)=2*192 = 364
a. Find the most general real-valued solution to the linear system of differential equations \( \overrightarrow{\boldsymbol{x}}^{\prime}=\left[\begin{array}{rr}-4 & -9 \\ 1 & -4\end{array}\right] \ove
The most general real-valued solution to the linear system of differential equations,[tex]\( \overrightarrow{\boldsymbol{x}}^{\prime}=\left[\begin{array}{rr}-4 & -9 \\ 1 & -4\end{array}\right] \overrightarrow{\boldsymbol{x}} \),[/tex] can be found by diagonalizing the coefficient matrix and using the exponential of the diagonal matrix.
To find the most general real-valued solution to the given linear system of differential equations, we start by finding the eigenvalues and eigenvectors of the coefficient matrix [tex]\(\left[\begin{array}{rr}-4 & -9 \\ 1 & -4\end{array}\right]\).[/tex]
Solving for the eigenvalues, we get:
[tex]\((-4-\lambda)(-4-\lambda) - (-9)(1) = 0\)\(\lambda^2 + 8\lambda + 7 = 0\)\((\lambda + 7)(\lambda + 1) = 0\)\(\lambda_1 = -7\) and \(\lambda_2 = -1\)[/tex]
Next, we find the corresponding eigenvectors:
For [tex]\(\lambda_1 = -7\):[/tex]
[tex]\(\left[\begin{array}{rr}-4 & -9 \\ 1 & -4\end{array}\right]\left[\begin{array}{r}x_1 \\ x_2\end{array}\right] = -7\left[\begin{array}{r}x_1 \\ x_2\end{array}\right]\)[/tex]
This leads to the equation:[tex]\(-4x_1 - 9x_2 = -7x_1\)[/tex], which simplifies to [tex]\(3x_1 + 9x_2 = 0\)[/tex]. Choosing[tex]\(x_2 = 1\),[/tex] we get the eigenvector [tex]\(\mathbf{v}_1 = \left[\begin{array}{r}3 \\ 1\end{array}\right]\).[/tex]
For[tex]\(\lambda_2 = -1\):\(\left[\begin{array}{rr}-4 & -9 \\ 1 & -4\end{array}\right]\left[\begin{array}{r}x_1 \\ x_2\end{array}\right] = -1\left[\begin{array}{r}x_1 \\ x_2\end{array}\right]\)[/tex]
This gives the equation:[tex]\(-4x_1 - 9x_2 = -x_1\),[/tex] which simplifies to[tex]\(3x_1 + 9x_2 = 0\).[/tex] Choosing [tex]\(x_2 = -1\)[/tex], we obtain the eigenvector [tex]\(\mathbf{v}_2 = \left[\begin{array}{r}-3 \\ 1\end{array}\right]\).[/tex]
Now, using the diagonalization formula, the general solution can be expressed as:
[tex]\(\overrightarrow{\boldsymbol{x}} = c_1e^{\lambda_1 t}\mathbf{v}_1 + c_2e^{\lambda_2 t}\mathbf{v}_2\)\(\overrightarrow{\boldsymbol{x}} = c_1e^{-7t}\left[\begin{array}{r}3 \\ 1\end{array}\right] + c_2e^{-t}\left[\begin{array}{r}-3 \\ 1\end{array}\right]\),[/tex]
where[tex]\(c_1\) and \(c_2\)[/tex] are constants.
Learn more about diagonal matrix here:
https://brainly.com/question/28217816
#SPJ11
Find the most general real-valued solution to the linear system of differential equations[tex]\( \overrightarrow{\boldsymbol{x}}^{\prime}=\left[\begin{array}{rr}-4 & -9 \\ 1 & -4\end{array}\right] \ove[/tex]
A tower 155 m high is situated at the top of a hill at a point 655 m down the hill the angle bet. The surface of the hill and the line of sight to the top of the tower is 12° 30'. Find the inclination of the hill to a horizontal plane.
The inclination of the hill to a horizontal plane is found to be 17.22° (approx).
Given:
Height of the tower, AB = 155m
Distance between the tower and a point on the hill, BC = 655m
Angle of depression from B to the foot of the tower, A = 12°30'
Let, the angle of inclination of the hill to a horizontal plane be x.
In ΔABC, we have:
tan A = AB/BC
⇒ tan 12°30' = 155/655
⇒ tan 12°30' = 0.2671
Now, consider the right-angled triangle ABP drawn below:
In right triangle ABP, we have:
tan x = BP/AP
⇒ tan x = BP/BC + CP
⇒ tan x = BP/BC + AB tan A
Here, we know AB and BC and we have just calculated tan A.
BP is the height of the hill from the horizontal plane, which we have to find.
Now, we have:
tan x = BP/BC + AB tan A
⇒ tan x = BP/655 + 155 × 0.2671
⇒ tan x = BP/655 + 41.1245
⇒ tan x = (BP + 655 × 41.1245)/655
⇒ BP + 655 × 41.1245 = 655 × tan x
⇒ BP = 655(tan x - 41.1245)
Thus, the angle of inclination of the hill to a horizontal plane is
x = arctan[BP/BC + AB tan A]
= arctan[(BP + 655 × 41.1245)/655].
Hence, the value of the inclination of the hill to a horizontal plane is 17.22° (approx).
Know more about the Angle of depression
https://brainly.com/question/17193804
#SPJ11
In an experiment, a group of college students was told that they were participating in a manual skill study. Half of the students were given a stack of money to count and the other half got a stack of blank pieces of paper. After the counting task, the participants were asked to dip their hands into bowls of very hot water (122°F) and rate how uncomfortable it was. Given the following data; Find the estimated error of the mean only. For counting Money: n1=10, M1-216, S51-216 For counting Paper: n2 =10, M2-60, SS2=383 I Please type the estimated error of the mean only in nearest hundredths place.
The estimated error of the mean only in nearest hundredths place is approximately 21.62.
To find the estimated error of the mean, we need to calculate the standard error for each group and then use the formula for the difference in means.
The formula for the standard error of the mean (SE) is:
SE = √((S²) / n)
where S is the sample standard deviation and n is the sample size.
For the group counting money:
n1 = 10 (sample size)
S1 = 216 (sample standard deviation)
SE1 = √((S1²) / n1)
= √((216²) / 10)
= √(46656 / 10)
= √(4665.6)
≈ 68.28
For the group counting paper:
n2 = 10 (sample size)
S2 = √(SS2 / (n2 - 1)) = √(383 / 9) ≈ 6.83 (sample standard deviation)
SE2 = √((S2²) / n2)
= √((6.83²) / 10)
= √(46.7089 / 10)
= √(4.67089)
≈ 2.16
Now, we can calculate the estimated error of the mean (EE) using the formula:
EE = √((SE1²) / n1 + (SE2²) / n2)
EE = √((68.28²) / 10 + (2.16²) / 10)
=√(4665.6384 / 10 + 4.6656 / 10)
= √(466.56384 + 0.46656)
=√(466.56384 + 0.46656)
= √(467.0304)
≈ 21.62
Therefore, the estimated error of the mean is approximately 21.62.
Learn more about standard error here:
https://brainly.com/question/32854773
#SPJ11
Identify the quadrant or quadrants for the angle \( \theta \) satisfying the given condition. \( \cot (\theta)>0 \) and \( \cos (\theta)
The quadrant or quadrants for the angle satisfying the given condition are the Quadrant 1 and Quadrant 3.
Given that cot(θ) > 0 and cos(θ) < 0.The range of cot(θ) is all real numbers except the odd multiples of π/2 and the range of cos(θ) is between -1 and 1. Therefore, the angle θ satisfies the given condition only if it lies in Quadrant 1 or Quadrant 3, since cot is positive and cosine is negative in these quadrants.
In Quadrant 1, all trigonometric functions are positive. Here, the reference angle, θr, is the same as the angle, θ, so cos(θ) is positive but cot(θ) is positive. Also, the opposite side of θr is equal to the adjacent side of θ, but the hypotenuse of θr is always smaller than that of θ.
In Quadrant 3, only tangent and cosecant are positive. Here, the reference angle, θr, is 180° − θ, so the sine and cosecant of θ are negative but the cotangent and cosine are positive. Also, the opposite side of θ is equal to the adjacent side of θr, but the hypotenuse of θ is always smaller than that of θr.
To know more about quadrants refer here:
https://brainly.com/question/26426112
#SPJ11
Find the dimensions of the rectangle with perimeter 1120 inches with the largest possible area. (For this problem, if necessary, assume that the length is the less than or equal to the width.) length = width = What is the maximum area? area =
The maximum area of the rectangle is 78,400 square inches.
Let's assume that the length of the rectangle is represented by L and the width is represented by W.
We know that the perimeter of a rectangle is given by the formula:
Perimeter = 2L + 2W
Given that the perimeter is 1120 inches, we can set up the equation:
2L + 2W = 1120
Dividing both sides of the equation by 2, we get:
L + W = 560
To maximize the area of the rectangle, we need to find the dimensions that satisfy the given perimeter constraint and maximize the product of length and width (area = L * W).
To do this, we can rewrite the equation above as:
L = 560 - W
Substituting this expression for L in the area equation, we have:
Area = (560 - W) * W
Expanding the equation, we get:
Area = 560W - W^2
To find the maximum area, we can differentiate the area equation with respect to W and set it equal to zero:
d(Area)/dW = 560 - 2W = 0
Solving for W, we have:
560 - 2W = 0
2W = 560
W = 280
Substituting this value back into the equation for L, we get:
L = 560 - W = 560 - 280 = 280
Therefore, the dimensions of the rectangle with the largest possible area are:
Length = Width = 280 inches
To find the maximum area, we substitute the values of L and W into the area equation:
Area = L * W = 280 * 280 = 78,400 square inches
Know more about rectanglehere;
https://brainly.com/question/15019502
#SPJ11