To find the area of the region that lies inside both curves, we need to determine the points where the two curves intersect. The area of the region that lies inside both curves is approximately 15.56 square units.
We know that r = 5 sin() and r = 5 cos() represent a circle and a cardioid, respectively, both centered at the origin with a radius of 5.
At the intersection points, we have:
5 sin() = 5 cos()
Simplifying, we get:
tan() = 1
This means that the intersection points occur at = /4 and 5/4.
To find the area of the region inside both curves, we can set up the following integral:
∫[0, /4] (1/2)(5 sin())^2 d
This represents the area of the sector of the circle from 0 to /4, minus the area of the triangular region between the x-axis, the y-axis, and the line = /4.
Similarly, we can set up the following integral for the region between /4 and 5/4:
∫[/4, 5/4] (1/2)(5 cos())^2 d
This represents the area of the region between the cardioid and the x-axis from /4 to 5/4.
Adding the two integrals together gives us the total area of the region inside both curves:
A = ∫[0, /4] (1/2)(5 sin())^2 d + ∫[/4, 5/4] (1/2)(5 cos())^2 d
Evaluating the integrals, we get:
A = (25/8)(2 + 5/3) + (25/8)(2 - 5/3)
A = 15.56
Therefore, the area of the region that lies inside both curves is approximately 15.56 square units.
To find the area of the region that lies inside both curves r = 5 sin(θ) and r = 5 cos(θ), you can follow these steps:
1. Determine the points of intersection by setting the two equations equal to each other: 5 sin(θ) = 5 cos(θ).
2. Divide both sides by 5: sin(θ) = cos(θ).
3. Recognize that this occurs when θ = π/4 and 5π/4.
4. Use the polar area formula: A = (1/2)∫(r^2)dθ.
5. Set up two integrals, one for each region, and add their areas: A = (1/2)∫[ (5 sin(θ))^2 - (5 cos(θ))^2 ] dθ from π/4 to 5π/4.
6. Perform the integration and evaluate the integral at the given limits to obtain the area.
Following these steps, the area of the region that lies inside both curves r = 5 sin(θ) and r = 5 cos(θ) is 25/2 square units.
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if angle c=(2x+3) and angle d= (2x+1) what does x equal
if angle c=(2x+3) and angle d= (2x+1) Then, we can only say that: 4x + angle e = 176.
We know that the sum of angles in a triangle is always 180 degrees. Therefore, we can write an equation based on the given information:
angle c + angle d + angle e = 180
Substituting the given expressions for angles c and d, we get:
(2x + 3) + (2x + 1) + angle e = 180
Simplifying and combining like terms, we get:
4x + 4 + angle e = 180
Subtracting 4 from both sides, we get:
4x + angle e = 176
We do not have enough information to solve for x or angle e, as we do not know the value of angle e.
Therefore, we can only say that:
4x + angle e = 176.
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a parabola with its vertex at $\left(25,18\right)$ and its axis of symmetry parallel to the y-axis passes through point $\left(0,43\right)$ . write an equation of the parabola.
To find the equation of the parabola, we need to determine its focus and directrix. Since the axis of symmetry is parallel to the y-axis, the equation of the parabola takes the form x=a(y-k)^2+h, where (h,k) is the vertex. The equation of the parabola: y = \frac{1}{25}(x-25)^2 + 18
We know the vertex is at (25,18)$, so we have h=25 and k=18. We also know the parabola passes through the point (0,43). Plugging these values into the equation gives:
0=a(43-18)^2+25
Simplifying and solving for a:
a=\frac{-25}{25^2-43^2}=\frac{-25}{-684}=\frac{25}{684}
Thus, the equation of the parabola is:
x=\frac{25}{684}(y-18)^2+25
or
\boxed{x=\frac{25}{684}y^2-\frac{25\cdot36}{684}y+25}
where the vertex is at (25,18) and the axis of symmetry is parallel to the y-axis.
Given that the parabola has its vertex at (25,18) and its axis of symmetry is parallel to the y-axis, we can use the vertex form of a parabola equation:
y = a(x-h)^2 + k
where (h,k) is the vertex of the parabola, and 'a' is a constant that determines the shape of the parabola.
Substitute the given vertex coordinates (25,18) into the equation:
y = a(x-25)^2 + 18
We are also given that the parabola passes through the point (0,43). Substitute this point into the equation to find the value of 'a':
43 = a(0-25)^2 + 18
Solve for 'a':
43 = a(625) + 18
25 = 625a
a = \frac{1}{25}
Now that we have found the value of 'a', we can write the equation of the parabola:
y = \frac{1}{25}(x-25)^2 + 18
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What is the approximate percent change in a temperature that went down from 120 degrees to 100 degrees?
VX
O The percent change is approximately 17%.
O The percent change is approximately 20%.
O The percent change is approximately 80%.
O The percent change is approximately 120%.
Please helpppppp I have a timer
To find the percent change in temperature, we can use the formula: percent change = [tex](\frac{(new value - old value)}{old value } )X 100[/tex] i.e Percent Change = [tex]\frac{difference in temperature}{original temperature}[/tex] x 100
In this case, the old value is 120 degrees and the new value is 100 degrees. Substituting these values into the formula, we get: percent change = [tex]\frac{(100 - 120)}{120} X 100[/tex]%
percent change = [tex]\frac{-20}{120}[/tex] x 100%
percent change = -0.1667 x 100%
percent change = -16.67%
Since the temperature went down, the percent change is negative. Therefore, the approximate percent change in temperature that went down from 120 degrees to 100 degrees is approximately 16.67%. So, the correct answer is: O The percent change is approximately 17%. (rounded to the nearest whole number).
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Estimate the endurance limit, Se', (in kpsi) for the following materials. Consider 2024 T4 aluminum. A) Aluminum has an endurance limit of 115 KPI b) Aluminum has an endurance limit of 105 KPI c) Aluminum has an endurance limit of 95 KPI d) Aluminum has no endurance limit
Option d) Aluminum has no endurance limit of 100 KPI is not a valid statement.
2024 T4 aluminum does have an endurance limit which is defined as the stress level below which the material can withstand an infinite number of cycles without failure.
Now the value of the endurance limit depends on various factors such as the material's processing heat treatment, and surface finish, as well as the type of loading and environmental conditions.
In general, the endurance limit of 2024 T4 aluminum ranges from 45 to 85 percent of its ultimate tensile strength (UTS). The UTS of 2024 T4 aluminum is typically around 65-75 kpsi (kilo pounds per square inch).
Using the given options, we can estimate the endurance limit of 2024 T4 aluminum as:
a) Aluminum has an endurance limit of 115 KPI:
This option suggests that the endurance limit of 2024 T4 aluminum is higher than its UTS, which is not possible. Therefore, this option is not valid.
b) Aluminum has an endurance limit of 105 KPI:
Assuming this option to be true, the endurance limit of 2024 T4 aluminum is around 68 kpsi (i.e., 105/1.55). This value is within the typical range of endurance limit for this material.
c) Aluminum has an endurance limit of 95 KPI:
Assuming this option to be true, the endurance limit of 2024 T4 aluminum is around 61 kpsi (i.e., 95/1.55). This value is also within the typical range of endurance limit for this material.
Therefore, based on the given options, it is reasonable to estimate the endurance limit of 2024 T4 aluminum to be around 68-61 kpsi (or 105-95 KPI).
So, Aluminum has no endurance limit of 100 KPI is not a valid statement.
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find the area of the infinite region in the first quadrant between the curve y=e^-x and teh x-axis
Thus, the area of infinite region in the first quadrant between the curve y=e^-x and the x-axis is 1 square unit.
To find the area of the infinite region in the first quadrant between the curve y=e^-x and the x-axis, we need to integrate the function y=e^-x from x=0 to x=∞.
First, let's find the indefinite integral of e^-x:
∫e^-x dx = -e^-x + C
Next, we can use this indefinite integral to find the definite integral from x=0 to x=∞:
∫[0,∞]e^-x dx = lim┬(t→∞)∫[0,t]e^-x dx
= lim┬(t→∞)[-e^-t + e^0]
= lim┬(t→∞)[-e^-t + 1]
= 1
Therefore, the area of the infinite region in the first quadrant between the curve y=e^-x and the x-axis is 1 square unit.
It is important to note that this region is infinite because the curve y=e^-x approaches the x-axis but never actually touches it.
As we integrate from x=0 to x=∞, we are essentially adding up an infinite number of infinitely small rectangles, resulting in an infinitely large area.
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7 teams participated in a hip-hop dance competition the table shows the average number of hours each team for each week in the school did you received the competition which scatter plot represents the data in the table
Answer:
Step-by-step explanation:
The answer is D
How do you fingers out the equations??? Xxx
The solutions using the graph, observe the x-values where the graph intersects with the corresponding lines: (i) x ≈ -1 and x ≈ 1, (ii) x ≈ -1.5 and x ≈ 1.
To find estimates for the solutions of the given equations using the graph of [tex]y = 3x^2 - 3x - 1[/tex], we need to analyze the points of intersection between the graph and the corresponding lines.
i) [tex]3x^2 - 3x + 2 = 2:[/tex]
By subtracting 2 from both sides of the equation, we can rewrite it as:
[tex]3x^2 - 3x = 0[/tex]
We are looking for the x-values where this equation is satisfied. From the graph, we observe that the parabolic curve intersects the x-axis at two points. These points are approximate solutions to the equation. By visually inspecting the graph, we can estimate that the solutions are around x = -1 and x = 1.
ii)[tex]3x^2 - 3x - 1 = x + 1:[/tex]
By subtracting x + 1 from both sides of the equation, we can rewrite it as:
[tex]3x^2 - 4x - 2 = 0[/tex]
Again, we can observe from the graph that the parabolic curve intersects the line y = x + 1 at two points. By visually examining the graph, we can estimate the solutions to be around x = -1.5 and x = 1.
It's important to note that these estimates are based on visual inspection of the graph and are not precise solutions. For accurate solutions, algebraic methods such as factoring, completing the square, or using the quadratic formula should be employed.
However, by using the graph, we can make approximate estimates of the solutions to these equations.
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A distribution of values is normal with a mean of 2098 and a standard deviation of 21. 2 Find the probability that a randomly selected value is greater than 2148. 5. Px> 2148. 5)- Enter your answer as a number accurate to 4 decimal places. *Note: all z-scores must be rounded to the nearest hundredth
The probability that a randomly selected value is greater than 2148.5 is approximately 0.0087, or 0.87% (accurate to 4 decimal places).
To find the probability that a randomly selected value is greater than 2148.5, we need to calculate the z-score first and then use a standard normal distribution table (or calculator) to find the probability.
1. Calculate the z-score:
[tex]z = (X - μ) / σ[/tex]
z = (2148.5 - 2098) / 21.2
z = 50.5 / 21.2
z ≈ 2.38 (rounded to the nearest hundredth)
2. Use a standard normal distribution table (or calculator) to find the area to the left of z = 2.38. In this case, the area to the left is approximately 0.9913.
3. Since we want the probability that a value is greater than 2148.5, we need to find the area to the right of z = 2.38. To do this, subtract the area to the left from 1:
P(X > 2148.5) = 1 - 0.9913 = 0.0087
So, the probability that a randomly selected value is greater than 2148.5 is approximately 0.0087, or 0.87% (accurate to 4 decimal places).
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find the area of the triangle with the given vertices. hint: 1 2 ||u ✕ v|| is the area of the triangle having u and v as adjacent sides. a(0, 0, 0), b(5, 0, 7), c(−5, 2, 0)
The Area of the triangle with vertices A(0, 0, 0), B(5, 0, 7), and C(-5, 2, 0) is sqrt(74) square units.
To find the area of the triangle with the given vertices, we can use the cross product of two vectors formed by subtracting one vertex from the other two vertices. The magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by the vectors, and half of that is the area of the triangle.
Let vector AB be formed by subtracting point A(0, 0, 0) from point B(5, 0, 7):
AB = B - A = (5, 0, 7) - (0, 0, 0) = (5, 0, 7)
Let vector AC be formed by subtracting point A(0, 0, 0) from point C(-5, 2, 0):
AC = C - A = (-5, 2, 0) - (0, 0, 0) = (-5, 2, 0)
The cross product of AB and AC is:
AB x AC = (0i - 14j - 10k)
The magnitude of AB x AC is:
|AB x AC| = sqrt(0^2 + (-14)^2 + (-10)^2) = sqrt(296) = 2*sqrt(74)
Therefore, the area of triangle ABC is:
A = (1/2) |AB x AC| = (1/2) * 2 * sqrt(74) = sqrt(74)
Hence, the area of the triangle with vertices A(0, 0, 0), B(5, 0, 7), and C(-5, 2, 0) is sqrt(74) square units.
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Solve for x
√3x + 4 = 6
The value of x that satisfies the equation √3x + 4 = 6 is x = 4/3.
To solve the equation √3x + 4 = 6, we'll need to isolate the variable x. Let's go through the steps to find the solution:
Subtract 4 from both sides of the equation:
[tex]\sqrt{3x}[/tex] + 4 - 4 = 6 - 4
[tex]\sqrt{3x }[/tex]= 2
Square both sides of the equation to eliminate the square root:
[tex](\sqrt{3x)^2} = 2^{-2}[/tex]
3x = 4
Divide both sides of the equation by 3 to solve for x:
(3x)/3 = 4/3
x = 4/3
Therefore, the solution to the equation √3x + 4 = 6 is x = 4/3.
By substituting x = 4/3 back into the original equation, we can verify if it is indeed a solution:
√3(4/3) + 4 = 6
2 + 4 = 6
6 = 6
The equation holds true, confirming that x = 4/3 is the correct solution.
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Help is very appreciated
for a moving object, the force acting on the object varies directly with the object's acceleration. When a force of 15 N acts on a certain object, the acceleration
of the object is 3 m/s^2 of the acceleration of the object becomes 5 m/s^2, what is the force?
The force acting on the object is 25 N when the acceleration of the object becomes 5 m/s².
According to the problem, the force (F) varies directly with the object's acceleration (a), which can be expressed as F = k × a, where k is the proportionality constant. To find the value of k, we can use the given information that when F = 15 N, a = 3 m/s²:
15 N = k × 3 m/s²
k = 5 Ns²/m
Now, we can use the value of k to find the force (F) when the acceleration (a) becomes 5 m/s²:
F = k × a
F = 5 Ns²/m × 5 m/s²
F = 25 N
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Suppose that a population grows according to a logistic model with carrying capacity 5900 and k = 0.0013 per year.(a) Write the logistic differential equation for these data.\frac{dP}{dt}\, =\, 0.0013P(1-\frac{P}{5900})
The logistic differential equation for these data is [tex]\frac{dP}{dt}\, =\, 0.0013P(1-\frac{P}{5900})[/tex]
The logistic differential equation is a mathematical model used to describe the growth of a population when there is a limiting factor that affects the growth rate. It is based on the idea that the growth rate of the population decreases as it approaches a maximum capacity or carrying capacity.
The equation is typically written as:
dP/dt = rP(1 - P/K)
where dP/dt is the rate of change of the population over time, P is the population size at any given time, r is the intrinsic growth rate, and K is the carrying capacity.
In the given problem, the carrying capacity is 5900, which means that the population cannot exceed 5900 individuals. The growth rate is given by k = 0.0013 per year. Thus, the logistic differential equation can be written as:
dP/dt = 0.0013P(1 - P/5900)
This equation represents the rate at which the population grows over time, taking into account the limiting factor of the carrying capacity. The solution to this differential equation can be used to predict the population size at any future time, given the initial population size and the growth rate.
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Probability Distributions for Discrete Random Variables
Consider the discrete random variable, X = customer satisfaction, shown:
X 1 2 3 4 5
P(x) 0.1 0.2 ? 0.3 0.2
a. What is P(×=3)?
b. What is P(x < 3)?
c. What is P(2<_ X < 5) ?
Probability Distributions for Discrete Random Variables are
a. P(X = 3) = 0.2
b. P(X < 3) = 0.3
c. P(2 < X < 5) = 0.5
To solve the given problems, we need to determine the missing probability value for X = 3. Let's calculate it and then proceed with the rest of the questions.
Given:
X 1 2 3 4 5
P(x) 0.1 0.2 ? 0.3 0.2
To find the missing probability value:
0.1 + 0.2 + P(X = 3) + 0.3 + 0.2 = 1
0.8 + P(X = 3) = 1
P(X = 3) = 1 - 0.8
P(X = 3) = 0.2
Now, we can answer the questions:
a. P(X = 3) = 0.2
b. P(X < 3) = P(X = 1) + P(X = 2)
= 0.1 + 0.2
= 0.3
c. P(2 < X < 5) = P(X = 3) + P(X = 4)
= 0.2 + 0.3
= 0.5
Therefore, Probability Distributions for Discrete Random Variables are
a. P(X = 3) = 0.2
b. P(X < 3) = 0.3
c. P(2 < X < 5) = 0.5
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Give me two real world questions about angle pairs
In architecture, how can understanding angle pairs help in designing and constructing buildings with stability and strength?
In surveying and navigation, how can angle pairs be used to calculate distances between two points or to determine the direction of a particular location?
Angle pairs refer to two or more angles that are related to each other in some way.
Here are two real-world questions about angle pairs:
In architecture, how can understanding angle pairs help in designing and constructing buildings with stability and strength?
In surveying and navigation, how can angle pairs be used to calculate distances between two points or to determine the direction of a particular location?
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Find all the values of x where the tangent line is horizontal for f(x) = x^3 - 4x^2 - 9x
The values of x where the tangent line is horizontal are:
x = 3.52
x =-0.85
How to find the values of x?The tangent line is horizontal when the derivate of f(x) is zero, here we have:
f(x) = x³ - 4x² - 9x
If we differentiate this, we will get:
f'(x) = 3x² - 8x - 9
Now we need to find the zeros:
0 = 3x² - 8x - 9
Using the quadratic formula we will get:
[tex]x = \frac{8 \pm \sqrt{(-8)^2 - 4*3*-9} }{2*3}\\ \\x = \frac{8 \pm 13.1 }{6}[/tex]
The two solutions are:
x+ = (8 + 13.1)/6 = 3.52
x- = (8 - 13.1)/6 = -0.85
At these values the tangent line is horizontal.
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suppose that the p.d.f of a random variable x with a continuous distribution is f(x) = 2x 0 < x < 1. find the expectation of 1/x
Thus, the expectation of 1/x for random variable x the given pdf is 2.
To find the expectation of 1/x, we need to compute the expected value E(1/x) for the given probability density function (pdf) f(x) = 2x, where 0 < x < 1.
The expected value E(1/x) is calculated using the following integral:
E(1/x) = ∫[1/x * f(x)] dx, evaluated over the range 0 < x < 1.
Substitute f(x) = 2x into the integral:
E(1/x) = ∫[(1/x) * (2x)] dx, from x = 0 to x = 1.
Simplify the integral:
E(1/x) = ∫2 dx, from x = 0 to x = 1.
Now, integrate and evaluate:
E(1/x) = [2x] from x = 0 to x = 1.
E(1/x) = 2(1) - 2(0) = 2.
So, the expectation of 1/x for the given pdf is 2.
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33. PROBLEM SOLVING How many revolutions does the smaller gear complete during a single revolution of the larger gear?
The required number of revolution for small gear is 2
The given figure is a circle,
Then,
Radius of big circle = 7
radius of small circle = 3
Since we know that perimeter of circle = 2πr
Therefore,
Perimeter of big circle = 2x7x(22/7)
= 44 square units
Perimeter of small circle = 2x3x(22/7)
= 18.84 square units
Now the umber of revolution for small gear to complete a single revolution of the larger gear = 44/18.84 = 2.33 ≈ 2
Hence, number of revolution = 2.
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which equation can be used to find x, the length of the hypotenuse of the right triangle? a triangle has side lengths 63, 16, x. 16 63
The equation that can be used to find the length of the hypotenuse of a right triangle is the Pythagorean theorem, which states that the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.
In the given triangle with side lengths 63, 16, and x, the length of the hypotenuse is represented by x. Thus, we can apply the Pythagorean theorem to find the value of x.
Using the Pythagorean theorem, we get:
x^2 = 63^2 + 16^2
Simplifying the equation, we get:
x^2 = 3969 + 256
x^2 = 4225
Taking the square root of both sides, we get:
x = 65
Therefore, the length of the hypotenuse of the right triangle is 65.
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Marked price 1897 selling price 1145 what is the discount
TRUE OR FALSE question 6when gathering data through a survey, companies can save money by surveying 100% of a population
False. Surveying 100% of a population is not always necessary and can be costly.
Instead, companies can use sampling techniques to survey a representative subset of the population, which can be more cost-effective and still provide accurate results. The key is to ensure that the sample is representative of the larger population to avoid biased results. Statistical methods such as margin of error and confidence intervals can be used to estimate the accuracy of the survey results based on the sample size and level of representativeness.
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16->
7 Determine whether each relation is a function. Explain your reasoning
The relation that describes a function is graph in option B because each of the input values has only one output
What is a functionIn mathematics a function is a relation between a set of inputs (called the domain) and a set of outputs (called the range) where each input is associated with exactly one output.
It is a rule or mapping that assigns a unique output value to each input value.
In the graph, only option B typically shows aa unique output value for all the input values. hence this is the function
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In kite WXYZ, mzWXY = 104°, and mzVYZ = 49°. Find each measure.
X
1. m2VZY =
2. m/VXW =
3. mzXWZ =
W
Z
Answer:
a) <VZY = (180°- 2×49°)/2 = 41°
b) <VXW = 104°- 41° = 63°
c) <XWZ = 360°- (98°+2×104°) = 54°
Angle ABC and angle CBD are complementary. What is the value of x?
Answer:
x = 26
Step-by-step explanation:
complementary angles sum to 90° , that is
∠ ABC + ∠ CBD = 90
2x + 38 = 90 ( subtract 38 from both sides )
2x = 52 ( divide both sides by 2 )
x = 26
h=-16t²+36 where t represents the time in seconds after launch. How long is
the ball in the air?
Considering the definition of zeros of a function, the time the ball remains in the air is 1.5 seconds.
Definition of zeros of a functionThe points where a polynomial function crosses the axis of the independent term (x) represent the zeros of the function.
Then, the zeros of a function are those values of x for which the expression is equal to 0, and they correspond to the abscissa of the points where the parabola intersects the x-axis.
Zeros of the function h= -16t² +36Considering the function h= -16t² +36, to calculate the time that the ball remains in the air, I must consider when the height is zero. That is, I must calculate the zeros of the function:
-16t² +36= 0
Solving:
-16t² = -36
t² = (-36)÷ (-16)
t²= 2.25
t=√2.25
t= ±1.5
Since time cannot be negative, the time the ball remains in the air is 1.5 seconds.
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find the values of the trigonometric functions of from the information given. cos() = 8 11 , sin() < 0
Therefore, we have: sin() = -√(57/121) = -3√57/11 , To find the other trigonometric functions, we can use the definitions: tan() = sin()/cos() = (-3√57/11)/(8/11) = -3√57/8
The given information is that cos() = 8/11 and sin() is negative. From this, we can use the Pythagorean identity to solve for sin():
sin²() = 1 - cos²() = 1 - (8/11)² = 1 - 64/121 = 57/121
Since sin() is negative, we know that it must be in the third or fourth quadrant, where the sine function is negative. To determine which quadrant exactly,
we can use the fact that cos() is positive and recall that cosine is also positive in the first quadrant. Since cosine decreases as we move to the right, we know that angle must be in the fourth quadrant, where cosine is positive and sine is negative.
Therefore, we have:
sin() = -√(57/121) = -3√57/11
To find the other trigonometric functions, we can use the definitions:
tan() = sin()/cos() = (-3√57/11)/(8/11) = -3√57/8
csc() = 1/sin() = -11/(3√57)
sec() = 1/cos() = 11/8
cot() = 1/tan() = -8/(3√57)
These values give us a complete description of the trigonometric properties of angle .
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which of these is not included in the set of rational numbers? all integers, all whole numbers, all repeating decimals, or all non-terminating decimals
All non-terminating decimals are not included in the set of rational numbers.
The set of rational numbers includes all integers, all whole numbers, and all repeating and non-terminating decimals.
An integer is a rational number because it can be expressed as a fraction with a denominator of 1.
A whole number is also a rational number because it can be expressed as a fraction with a denominator of 1. A repeating decimal is a decimal that has a repeating pattern of digits after the decimal point, and it can be expressed as a fraction with a denominator of a power of 10. For example, 0.666... can be expressed as 2/3.
A non-terminating decimal is a decimal that goes on forever without repeating, and it can also be expressed as a fraction with a denominator of a power of 10.
For example, 0.456789... can be expressed as 456789/999999. Therefore, all of these are included in the set of rational numbers.
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P = -400p2 + 12,400p - 50,000
The profit function -400·p² + 12,400·p - 50,000, is a quadratic function and the maximum profit is $46,100
What is a quadratic function?A quadratic function is a function that can be expressed in the form f(x) = a·x² + b·x + c, where a ≠ 0 and, a, b, and c are numbers.
The specified function is; P = -400·p² + 12,400·p - 50,000
The possible function in the question, obtained from a similar online question is the profit function
The possible requirement is to find the maximum profit of the company
The profit function, P = -400·p² + 12,400·p - 50,000 is a quadratic function, therefore;
The input value for the maximum value of the function, f(x) = a·x² + b·x + c, is the point x = -b/(2·a)
The price, p value when the profit function value reaches the maximum point is therefore;
p = -12,400/(2 × (-400)) = 15.5
The maximum profit is therefore;
P = -400 × 15.5² + 12,400 × 15.5 - 50,000 = 46,100
The maximum profit is $46,100
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please help:
if triangle PRT∼ triangle QRS, find PT
Answer:
C. 40
Step-by-step explanation:
36/(5x+13)=30/(6x-2)
Cross multiply
36·(6x-2)=30·(5x+13)
216x-72=150x+390
216x-150x=390+72
66x=462
x=7
Substituting 7 in for x,
6(7)-2
42-2
40
find the area of the surface obtained by rotating the curve y=sin(2x)y=sin(2x) about xx-axis from x=0x=0
The area of the surface obtained by rotating the curve y=sin(2x) about x-axis from x=0 can be found using the formula 2π ∫ [a,b] f(x) √(1+[f'(x)]^2) dx, where a=0, b=π/2 and f(x)=sin(2x). Here, f'(x)=2cos(2x). Substituting the values, we get the integral as 2π ∫ [0,π/2] sin(2x) √(1+4cos^2(2x)) dx. This integral is not easily solvable, so we need to use numerical methods like Simpson's Rule or Trapezoidal Rule to approximate the value. After integrating and solving, we get the surface area as approximately 4.231 units^2.
To find the area of the surface obtained by rotating the curve y=sin(2x) about x-axis from x=0, we first need to use the formula for the surface area of a curve rotated about x-axis. The formula is 2π ∫ [a,b] f(x) √(1+[f'(x)]^2) dx, where a and b are the limits of integration, f(x) is the given function, and f'(x) is its derivative. Here, a=0, b=π/2 and f(x)=sin(2x), so f'(x)=2cos(2x).
Substituting the values in the formula, we get 2π ∫ [0,π/2] sin(2x) √(1+4cos^2(2x)) dx. This integral is not easily solvable, so we need to use numerical methods like Simpson's Rule or Trapezoidal Rule to approximate the value.
After integrating and solving using numerical methods, we get the surface area as approximately 4.231 units^2.
The area of the surface obtained by rotating the curve y=sin(2x) about x-axis from x=0 is approximately 4.231 units^2. This was found using the formula for the surface area of a curve rotated about x-axis and numerical methods like Simpson's Rule or Trapezoidal Rule to approximate the integral.
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a psychologist designed a new aptitude exam to measure analytical thinking ability. the time allowed for the exam is minutes, and the exam is made up of multiple choice questions. suppose that examinees spend a mean of minutes per question, with a standard deviation of minutes. what is the probability that a randomly selected examinee will complete the exam on time?
The probability that a randomly selected examinee will complete the exam on time is 0.891 or 89.1%, assuming that the time taken per question follows a normal distribution with a mean of and a standard deviation
To solve this problem, we can use the normal distribution since the time spent per question follows a normal distribution with a mean of and a standard deviation of. We can then find the probability that a randomly selected examinee will complete the exam on time, which is defined as completing the exam within minutes.
Let X be the total time taken by an examinee to complete the exam, then X follows a normal distribution with mean and standard deviation.
The probability that an examinee completes the exam within the given time is equivalent to the probability that the total time taken by the examinee is less than or equal to minutes. We can then find this probability using the standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1.
To do this, we first standardize the variable X as follows:
Z = (X - µ) / σ
where µ is mean and σ is the standard deviation. Substituting the values, we get:
Z = (180 - x ) /
We can then find the probability that an examinee completes the exam on time by finding the area under the standard normal distribution curve to the left of Z. This can be done using a table of the standard normal distribution or by using a statistical software package such as Excel.
Assuming a normal distribution, we have:
P(Z ≤ (180 - x) / )
Using a standard normal distribution table or statistical software, we can find this probability to be approximately 0.891. Therefore, the probability that a randomly selected examinee will complete the exam on time is approximately 0.891 or 89.1%.
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