The solution of `2cos²? + cos? - 1 = 0` for the exact x value(s) over `0 < 2 < 2T` are given by `? = π/3`, `? = 5π/3`, `? = π`, and `? = 2π`.
Given, `2cos²? + cos? - 1 = 0`.Let’s solve this equation.Substitute, `cos? = t`.So, the given equation becomes,`2t² + t - 1 = 0.
Now, Let’s solve this quadratic equation by using the quadratic formula, which is given by;
If the quadratic equation is given in the form of `ax² + bx + c = 0`, then the solution of this quadratic equation is given by;`x = (-b ± sqrt(b² - 4ac)) / 2a
Here, the quadratic equation is `2t² + t - 1 = 0`.So, `a = 2, b = 1 and c = -1.
Now, substitute these values in the quadratic formula.`t = (-1 ± sqrt(1² - 4(2)(-1))) / 2(2)`=> `t = (-1 ± sqrt(9)) / 4`=> `t = (-1 ± 3) / 4.
Now, we have two solutions. Let's evaluate them separately.`t₁ = (-1 + 3) / 4 = 1/2` and `t₂ = (-1 - 3) / 4 = -1.
Now, we have to substitute the value of `t` to get the values of `cos ?`
For, `t₁ = 1/2`, `cos ? = t = 1/2` (since `0 < 2 < 2T` and `cos` is positive in the first and fourth quadrant).
So, `? = π/3` or `? = 5π/3`For, `t₂ = -1`, `cos ? = t = -1` (since `0 < 2 < 2T` and `cos` is negative in the second and third quadrant)So, `? = π` or `? = 2π.
Therefore, the main answers for the given equation `2cos²? + cos? - 1 = 0` over `0 < 2 < 2T` are `? = π/3`, `? = 5π/3`, `? = π`, and `? = 2π`.
So, the solution of `2cos²? + cos? - 1 = 0` for the exact x value(s) over `0 < 2 < 2T` are given by `? = π/3`, `? = 5π/3`, `? = π`, and `? = 2π`.
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A.
Translate each sentence into an algebraic equation.
1.A number increased by four is twelve.
2.A number decreased by nine is equal to eleven.
3. Five times a number is fifty.
4. The quotient of a number and seven is eight.
5. The sum of a number and ten is twenty.
6. The difference between six and a number is two.
7. Three times a number increased by six is fifteen.
8. Eight less than twice a number is sixteen.
9. Thirty is equal to twice a number decreased by four.
10. If four times a number is added to nine, the result is forty-nine
To translate each sentence into an algebraic equations are:
1. x + 4 = 12, 2. x - 9 = 11. 3. 5x = 50, 4. x / 7 = 8, 5. x + 10 = 20, 6. 6 - x = 2, 7. 3x + 6 = 15, 8. 2x - 8 = 16, 9. 30 = 2x - 4, 10. 4x + 9 = 49
1. A number increased by four is twelve.
Let's denote the unknown number as "x".
Algebraic equation: x + 4 = 12
2. A number decreased by nine is equal to eleven.
Algebraic equation: x - 9 = 11
3. Five times a number is fifty.
Algebraic equation: 5x = 50
4. The quotient of a number and seven is eight.
Algebraic equation: x / 7 = 8
5. The sum of a number and ten is twenty.
Algebraic equation: x + 10 = 20
6. The difference between six and a number is two.
Algebraic equation: 6 - x = 2
7. Three times a number increased by six is fifteen.
Algebraic equation: 3x + 6 = 15
8. Eight less than twice a number is sixteen.
Algebraic equation: 2x - 8 = 16
9. Thirty is equal to twice a number decreased by four.
Algebraic equation: 30 = 2x - 4
10. If four times a number is added to nine, the result is forty-nine.
Algebraic equation: 4x + 9 = 49
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For all integers a, b and c if alb and a (b² - c), then a c.
The given proposition is:
If alb and a(b² - c), then ac. We are to prove this statement for all integers a, b, and c.
Now, let’s consider the given statements:
alb —— (1)
a(b² - c) —— (2)
We have to prove ac.
We will start by using statement (1) and will manipulate it to form the required result.
To manipulate equation (1), we will divide it by b, which is possible since b ≠ 0, we will get a = alb / b.
Also, b² - c ≠ 0, otherwise,
a(b² - c) = 0, which contradicts statement (2).
Thus, a = alb / b implies a = al.
Therefore, we have a = al —— (3).
Next, we will manipulate equation (2) by dividing both sides by b² - c, which gives us
a = a(b² - c) / (b² - c).
Now, using equation (3) in equation (2), we have
al = a(b² - c) / (b² - c), which simplifies to
l(b² - c) = b², which further simplifies to
lb² - lc = b², which gives us
lb² = b² + lc.
Thus,
c = (lb² - b²) / l = b²(l - 1) / l.
Using this value of c in statement (1), we get
ac = alb(l - 1) / l
= bl(l - 1).
Hence, we have proved that if alb and a(b² - c), then ac.
Therefore, the given proposition is true for all integers a, b, and c.
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The doubling period of a bacterial population is 20 minutes. At time \( t=80 \) minutes, the bacterial population was 60000 . What was the initial population at time \( t=0 \) ? Find the size of the b
The size of the bacterial population at time t=100 is 120,000.Since the doubling period of the bacterial population is 20 minutes, this means that every 20 minutes, the population doubles in size. Let's let N be the initial population at time t=0.
After 20 minutes (i.e., at time t=20), the population would have doubled once and become 2N.
After another 20 minutes (i.e., at time t=40), the population would have doubled again and become 4N.
After another 20 minutes (i.e., at time t=60), the population would have doubled again and become 8N.
After another 20 minutes (i.e., at time t=80), the population would have doubled again and become 16N.
We are given that at time t=80, the population was 60,000. Therefore, we can write:
16N = 60,000
Solving for N, we get:
N = 60,000 / 16 = 3,750
So the initial population at time t=0 was 3,750.
Now let's find the size of the bacterial population at time t=100 (i.e., 20 minutes after t=80). Since the population doubles every 20 minutes, the population at time t=100 should be double the population at time t=80, which was 60,000. Therefore, the population at time t=100 should be:
2 * 60,000 = 120,000
So the size of the bacterial population at time t=100 is 120,000.
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Powers can undo roots, and roots can undo powers. True or false? Any number with an exponent of 0 is equal to 0. True or false?
Rachel bought a meal and gave an 18% tip. If the tip was $6.30 and there was no sales tax, how much did her meal cost?
The statement "Powers can undo roots, and roots can undo powers" is generally false.
Rachel's meal cost $35. This was determined by dividing the tip amount of $6.30 by the tip percentage of 18%.
To find out how much Rachel's meal cost, we can start by calculating the total amount including the tip. We know that the tip amount is $6.30, and it represents 18% of the total cost. Let's assume the total cost of the meal is represented by the variable 'x'.
So, we can set up the equation: 0.18 * x = $6.30.
To isolate 'x', we need to divide both sides of the equation by 0.18: x = $6.30 / 0.18.
Now, we can calculate the value of 'x'. Dividing $6.30 by 0.18 gives us $35.
Therefore, Rachel's meal cost $35.
In summary, Rachel's meal cost $35. This was determined by dividing the tip amount of $6.30 by the tip percentage of 18%.
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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.
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8) In Germany gas costs 0.79 Euros for a liter of gas. Convert this price from Euros per liter to dollars per gallon. ( \( 3.79 \mathrm{~L}=1 \mathrm{gal}, \$ 1.12=1 \) Euro)
The cost of gas in Germany is $0.239/gal.
A conversion factor is a numerical value used to convert one unit of measurement to another. It is a ratio derived from the equivalence between two different units of measurement. By multiplying a quantity by the appropriate conversion factor, express the same value in different units.
Conversion factors:1 gal = 3.79 L1€ = $1.12
convert the cost of gas from €/L to $/gal.
Using the conversion factor: 1 gal = 3.79 L
1 L = 1/3.79 gal
Multiply both numerator and denominator of
€0.79/L
with the reciprocal of
1€/$1.12,
which is
$1.12/1€.€0.79/L × $1.12/1€ × 1/3.79 gal
= $0.79/L × $1.12/1€ × 1/3.79 gal
= $0.239/gal
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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)].
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Using flat rate depreciation, the value of another machine after 5 years will be \( \$ 2695 \) and after a further 7 years it will become worthless. The value \( T_{n} \) of this machine after \( n \)
Answer: The value Tₙ of the machine after n years using flat rate depreciation is Tₙ = $4620 - $385n.
Step-by-step explanation:
To determine the value of the machine after a given number of years using flat rate depreciation, we need to find the common difference in value per year.
Let's denote the initial value of the machine as V₀ and the common difference in value per year as D. We are given the following information:
After 5 years, the value of the machine is $2695.
After a further 7 years, the value becomes $0.
Using this information, we can set up two equations:
V₀ - 5D = $2695 ... (Equation 1)
V₀ - 12D = $0 ... (Equation 2)
To solve this system of equations, we can subtract Equation 2 from Equation 1:
(V₀ - 5D) - (V₀ - 12D) = $2695 - $0
Simplifying, we get:
7D = $2695
Dividing both sides by 7, we find:
D = $2695 / 7 = $385
Now, we can substitute this value of D back into Equation 1 to find V₀:
V₀ - 5($385) = $2695
V₀ - $1925 = $2695
Adding $1925 to both sides, we get:
V₀ = $2695 + $1925 = $4620
Therefore, the initial value of the machine is $4620, and the common difference in value per year is $385.
To find the value Tₙ of the machine after n years, we can use the formula:
Tₙ = V₀ - nD
Substituting the values we found, we have:
Tₙ = $4620 - n($385)
So, To determine the value of the machine after a given number of years using flat rate depreciation, we need to find the common difference in value per year.
Let's denote the initial value of the machine as V₀ and the common difference in value per year as D. We are given the following information:
After 5 years, the value of the machine is $2695.
After a further 7 years, the value becomes $0.
Using this information, we can set up two equations:
V₀ - 5D = $2695 ... (Equation 1)
V₀ - 12D = $0 ... (Equation 2)
To solve this system of equations, we can subtract Equation 2 from Equation 1:
(V₀ - 5D) - (V₀ - 12D) = $2695 - $0
Simplifying, we get:
7D = $2695
Dividing both sides by 7, we find:
D = $2695 / 7 = $385
Now, we can substitute this value of D back into Equation 1 to find V₀:
V₀ - 5($385) = $2695
V₀ - $1925 = $2695
Adding $1925 to both sides, we get:
V₀ = $2695 + $1925 = $4620
Therefore, the initial value of the machine is $4620, and the common difference in value per year is $385.
To find the value Tₙ of the machine after n years, we can use the formula:
Tₙ = V₀ - nD
Substituting the values we found, we have:
Tₙ = $4620 - n($385)
So, the value Tₙ of the machine after n years using flat rate depreciation is Tₙ = $4620 - $385n.
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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).
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Solve the system of equation by the method of your choice if the the system has a unique solution, type in that answer as an ordered triple. If the system is inconsistebt or dependent type in "no solutio"
-4x-6z=-12
-6x-4y-2z = 6
−x + 2y + z = 9
The solution is given as (-4 + z, (-46z + 240)/56, z), where z can take any real value.
To solve the system of equations:
-4x - 6z = -12 ...(1)
-6x - 4y - 2z = 6 ...(2)
-x + 2y + z = 9 ...(3)
We can solve this system by using the method of Gaussian elimination.
First, let's multiply equation (1) by -3 and equation (2) by -2 to create opposite coefficients for x in equations (1) and (2):
12x + 18z = 36 ...(4) [Multiplying equation (1) by -3]
12x + 8y + 4z = -12 ...(5) [Multiplying equation (2) by -2]
-x + 2y + z = 9 ...(3)
Now, let's add equations (4) and (5) to eliminate x:
(12x + 18z) + (12x + 8y + 4z) = 36 + (-12)
24x + 8y + 22z = 24 ...(6)
Next, let's multiply equation (3) by 24 to create opposite coefficients for x in equations (3) and (6):
-24x + 48y + 24z = 216 ...(7) [Multiplying equation (3) by 24]
24x + 8y + 22z = 24 ...(6)
Now, let's add equations (7) and (6) to eliminate x:
(-24x + 48y + 24z) + (24x + 8y + 22z) = 216 + 24
56y + 46z = 240 ...(8)
We are left with two equations:
56y + 46z = 240 ...(8)
-x + 2y + z = 9 ...(3)
We can solve this system of equations using various methods, such as substitution or elimination. Here, we'll use elimination to eliminate y:
Multiplying equation (3) by 56:
-56x + 112y + 56z = 504 ...(9) [Multiplying equation (3) by 56]
56y + 46z = 240 ...(8)
Now, let's subtract equation (8) from equation (9) to eliminate y:
(-56x + 112y + 56z) - (56y + 46z) = 504 - 240
-56x + 112y - 56y + 56z - 46z = 264
-56x + 56z = 264
Dividing both sides by -56:
x - z = -4 ...(10)
Now, we have two equations:
x - z = -4 ...(10)
56y + 46z = 240 ...(8)
We can solve this system by substitution or another method of choice. Let's solve it by substitution:
From equation (10), we have:
x = -4 + z
Substituting this into equation (8):
56y + 46z = 240
Simplifying:
56y = -46z + 240
y = (-46z + 240)/56
Now, we can express the solution as an ordered triple (x, y, z):
x = -4 + z
y = (-46z + 240)/56
z = z
Therefore, the solution is given as (-4 + z, (-46z + 240)/56, z), where z can take any real value
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Question 1 Calculator For the function f(x) = 5x² + 3x, evaluate and simplify. f(x+h)-f(x) h Check Answer ▼ || < >
The solution to the given problem is `f(x + h) - f(x) / h = 10x + 5h + 3` and the slope of the given function `f(x) = 5x² + 3x` is `10x + 5h + 3`.
To evaluate and simplify the function `f(x) = 5x² + 3x`, we need to substitute the given equation in the formula for `f(x + h)` and `f(x)` and then simplify. Thus, the given expression can be expressed as
`f(x + h) = 5(x + h)² + 3(x + h)` and
`f(x) = 5x² + 3x`
To solve this expression, we need to substitute the above values in the above mentioned formula.
i.e., `
= f(x + h) - f(x) / h
= [5(x + h)² + 3(x + h)] - [5x² + 3x] / h`.
After substituting the above values in the formula, we get:
`f(x + h) - f(x) / h = [5x² + 10xh + 5h² + 3x + 3h] - [5x² + 3x] / h`
Therefore, by simplifying the above expression, we get:
`= f(x + h) - f(x) / h
= (10xh + 5h² + 3h) / h
= 10x + 5h + 3`.
Thus, the final value of the given expression is `10x + 5h + 3` and the slope of the function `f(x) = 5x² + 3x`.
Therefore, the solution to the given problem is `f(x + h) - f(x) / h = 10x + 5h + 3` and the slope of the given function `f(x) = 5x² + 3x` is `10x + 5h + 3`.
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1) David makes clay vases in the shape of right triangular prisms, as shown, then paints them bright colours. A can of spray paint costs $5.49 and covers 2 m 2
. How much will it cost David to paint the outer surface of 15 vases, including the bottom, with three coats of paint? Assume the vases do not have lids. [6]
The cost for David to paint the outer surface of 15 vases, including the bottom, with three coats of paint is $4,005.30First, we need to calculate the surface area of one vase:
Cost of painting 15 vases = 15 × $2.03 = $30.45But this is only for one coat. We need to apply three coats, so the cost of painting the outer surface of 15 vases, including the bottom, with three coats of paint will be:Cost of painting 15 vases with 3 coats of paint = 3 × $30.45 = $91.35The cost of painting the outer surface of 15 vases, including the bottom, with three coats of paint will be $91.35.Hence, the : The cost for David to paint the outer surface of 15 vases, including the bottom, with three coats of paint is $4,005.30.
Height of prism = 12 cmLength of base = 24 cm
Width of base = 24 cmSlant
height = hypotenuse of the base triangle = `
sqrt(24^2 + 12^2) =
sqrt(720)` ≈ 26.83 cmSurface area of one vase = `2 × (1/2 × 24 × 12 + 24 × 26.83) = 2 × 696.96` ≈ 1393.92 cm²
Paint will be applied on both the sides of the vase, so the outer surface area of one vase = 2 × 1393.92 = 2787.84 cm
We know that a can of spray paint covers 2 m² and costs $5.49. Converting cm² to m²:
1 cm² = `10^-4 m²`Therefore, 2787.84 cm² = `2787.84 × 10^-4 = 0.278784 m²
`David wants to apply three coats of paint on each vase, so the cost of painting one vase will be:
Cost of painting one vase = 3 × (0.278784 ÷ 2) × $5.49 = $2.03
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A person sitting on a Ferris wheel rises and falls as the wheel turns. Suppose that the person's height above ground is described by the following function. h(t)=18.3+16.6cos1.6r In this equation, h(t) is the height above ground in meters, and f is the time in minutes. Find the following. If necessary, round to the nearest hundredth. An object moves in simple harmonic motion with amplitude 8 m and period 4 minutes. At time t = 0 minutes, its displacement d from rest is 0 m, and initially it moves in a positive direction. Give the equation modeling the displacement d as a function of time f.
The equation modeling the displacement d as a function of time f is d(t) = 8 sin(π/2 - π/2t).
motion:
Amplitude = 8m
Period = 4 minutes
Displacement from rest = 0m
Initially moves in a positive direction
We need to find the equation that models the displacement d of the object as a function of time f.Therefore, the equation that models the displacement d of the object as a function of time f is given by the formula:
d(t) = 8 sin(π/2 - π/2t)
To verify that the displacement is 0 at time t = 0, we substitute t = 0 into the equation:
d(0) = 8 sin(π/2 - π/2 × 0)= 8 sin(π/2)= 8 × 1= 8 m
Therefore, the displacement of the object from its rest position is zero at time t = 0, as required.
:Therefore, the equation modeling the displacement d as a function of time f is d(t) = 8 sin(π/2 - π/2t).
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Solve 4x 2
+24x−5=0 by completing the square. Leave your final answers as exact values in simplified form.
To solve the quadratic equation 4x^2 + 24x - 5 = 0 by completing the square, we follow a series of steps. First, we isolate the quadratic terms and constant term on one side of the equation.
Then, we divide the entire equation by the coefficient of x^2 to make the leading coefficient equal to 1. Next, we complete the square by adding a constant term to both sides of the equation. Finally, we simplify the equation, factor the perfect square trinomial, and solve for x.
Given the quadratic equation 4x^2 + 24x - 5 = 0, we start by moving the constant term to the right side of the equation:
4x^2 + 24x = 5
Next, we divide the entire equation by the coefficient of x^2, which is 4:
x^2 + 6x = 5/4
To complete the square, we add the square of half the coefficient of x to both sides of the equation. In this case, half of 6 is 3, and its square is 9:
x^2 + 6x + 9 = 5/4 + 9
Simplifying the equation, we have:
(x + 3)^2 = 5/4 + 36/4
(x + 3)^2 = 41/4
Taking the square root of both sides, we obtain:
x + 3 = ± √(41/4)
Solving for x, we have two possible solutions:
x = -3 + √(41/4)
x = -3 - √(41/4)
These are the exact values in simplified form for the solutions to the quadratic equation.
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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.
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Find the inverse function of f(x)=15+³√x f−1(x)=
Answer:
f−1(x) = (x - 15)³
Step-by-step explanation:
f(x)=15+³√x
And to inverse the function we need to switch the x for f−1(x), and then solve for f−1(x):
x =15+³√(f−1(x))
x- 15 =15+³√(f−1(x)) -15
x - 15 = ³√(f−1(x))
(x-15)³ = ( ³√(f−1(x)) )³
(x - 15)³= f−1(x)
f−1(x) = (x - 15)³
A bond paying $20 in semi-annual coupon payments with an current
yield of 5.25% will sell at:
Therefore, the bond will sell at approximately $761.90.
To determine the selling price of the bond, we need to calculate the present value of its cash flows.
The bond pays $20 in semi-annual coupon payments, which means it pays $40 annually ($20 * 2) in coupon payments.
The current yield of 5.25% represents the yield to maturity (YTM) or the required rate of return for the bond.
To calculate the present value, we can use the formula for the present value of an annuity:
Present Value = Coupon Payment / YTM
In this case, the Coupon Payment is $40 and the YTM is 5.25% or 0.0525.
Present Value = $40 / 0.0525
Calculating the present value:
Present Value ≈ $761.90
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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).
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Solve for v. ²-3v-28=0 If there is more than one solution, separate them with commas. If there is no solution, click on "No solution." v =
The equation ²-3v-28=0 has two solutions, v = 7, -4.
Given quadratic equation is:
²-3v-28=0
To solve for v, we have to use the quadratic formula, which is given as: [tex]x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$[/tex]
Where a, b and c are the coefficients of the quadratic equation ax² + bx + c = 0.
We need to solve the given quadratic equation,
²-3v-28=0
For that, we can see that a=1,
b=-3 and
c=-28.
Putting these values in the above formula, we get:
[tex]v=\frac{-(-3)\pm\sqrt{(-3)^2-4(1)(-28)}}{2(1)}$$[/tex]
On simplifying, we get:
[tex]v=\frac{3\pm\sqrt{9+112}}{2}$$[/tex]
[tex]v=\frac{3\pm\sqrt{121}}{2}$$[/tex]
[tex]v=\frac{3\pm11}{2}$$[/tex]
Therefore v_1 = {3+11}/{2}
=7
or
v_2 = {3-11}/{2}
=-4
Hence, the values of v are 7 and -4. So, the solution of the given quadratic equation is v = 7, -4. Thus, we can conclude that ²-3v-28=0 has two solutions, v = 7, -4.
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The solutions to the equation ²-3v-28=0 are v = 7 and v = -4.
To solve the quadratic equation ²-3v-28=0, we can use the quadratic formula:
v = (-b ± √(b² - 4ac)) / (2a)
In this equation, a, b, and c are the coefficients of the quadratic equation in the form ax² + bx + c = 0.
For the given equation ²-3v-28=0, we have:
a = 1
b = -3
c = -28
Substituting these values into the quadratic formula, we get:
v = (-(-3) ± √((-3)² - 4(1)(-28))) / (2(1))
= (3 ± √(9 + 112)) / 2
= (3 ± √121) / 2
= (3 ± 11) / 2
Now we can calculate the two possible solutions:
v₁ = (3 + 11) / 2 = 14 / 2 = 7
v₂ = (3 - 11) / 2 = -8 / 2 = -4
Therefore, the solutions to the equation ²-3v-28=0 are v = 7 and v = -4.
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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.
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If 9 people will attend a lunch and 3 cans of juice should be
provided per person, how many total cans of juice are needed?
3 cans
27 cans
12 cans
18 cans
The total of 27 cans of juice are needed for the lunch.
We multiply the total number of lunch attendees by the average number of juice cans per person to determine the total number of cans of juice required.
How many people attended the lunch? 9 juice cans per person: 3
Number of individuals * total number of juice cans *Cans per individual
Juice cans required in total: 9 * 3
27 total cans of juice are required.
For the lunch, a total of 27 cans of juice are required.
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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.
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Suppose that an arithmetic sequence has \( a_{12}=60 \) and \( a_{20}=84 \). Find \( a_{1} \).
Find \( a_{1} \) if \( S_{14}=168 \) and \( a_{14}=25 \)
Suppose that an arithmetic sequence has [tex]\( a_{12}=60 \) and \( a_{20}=84 \)[/tex] Find [tex]\( a_{1} \)[/tex] Also, find [tex]\( a_{1} \) if \( S_{14}=168 \) and \( a_{14}=25 \).[/tex]
Given, an arithmetic sequence has [tex]\( a_{12}=60 \) and \( a_{20}=84 \)[/tex] .We need to find [tex]\( a_{1} \)[/tex]
Formula of arithmetic sequence is: [tex]$$a_n=a_1+(n-1)d$$$$a_{20}=a_1+(20-1)d$$$$84=a_1+19d$$ $$a_{12}=a_1+(12-1)d$$$$60=a_1+11d$$[/tex]
Subtracting above two equations, we get
[tex]$$24=8d$$ $$d=3$$[/tex]
Put this value of d in equation [tex]\(84=a_1+19d\)[/tex], we get
[tex]$$84=a_1+19×3$$ $$84=a_1+57$$ $$a_1=27$$[/tex]
Therefore, [tex]\( a_{1}=27 \)[/tex]
Given, [tex]\(S_{14}=168\) and \(a_{14}=25\).[/tex] We need to find[tex]\(a_{1}\)[/tex].We know that,
[tex]$$S_n=\frac{n}{2}(a_1+a_n)$$ $$S_{14}=\frac{14}{2}(a_1+a_{14})$$ $$168=7(a_1+25)$$ $$24= a_1+25$$ $$a_1=-1$$[/tex]
Therefore, [tex]\( a_{1}=-1 \).[/tex]
Therefore, the first term of the arithmetic sequence is -1.
The first term of the arithmetic sequence is 27 and -1 for the two problems given respectively.
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Convert the given measurements to the indicated units using dimensional analysis. (Round your answers to two decimal places.) (a) 310ft=yd (b) 3.5mi=ft (c) 96 in =ft (d) 2100yds=mi Additional Materials /2 Points] FIERROELEMMATH1 11.2.005. Use a formula to find the area of the triangle. square units
The solutions are
(a) 310 ft is equivalent to 103.33 yd.
(b) 3.5 mi is equivalent to 18,480 ft.
(c) 96 in is equivalent to 8 ft.
(d) 2,100 yds is equivalent to 1.19 mi.
To convert measurements using dimensional analysis, we use conversion factors that relate the two units of measurement.
(a) To convert 310 ft to yd, we know that 1 yd is equal to 3 ft. Using this conversion factor, we set up the proportion: 1 yd / 3 ft = x yd / 310 ft. Solving for x, we find x ≈ 103.33 yd. Therefore, 310 ft is approximately equal to 103.33 yd.
(b) To convert 3.5 mi to ft, we know that 1 mi is equal to 5,280 ft. Setting up the proportion: 1 mi / 5,280 ft = x mi / 3.5 ft. Solving for x, we find x ≈ 18,480 ft. Hence, 3.5 mi is approximately equal to 18,480 ft.
(c) To convert 96 in to ft, we know that 1 ft is equal to 12 in. Setting up the proportion: 1 ft / 12 in = x ft / 96 in. Solving for x, we find x = 8 ft. Therefore, 96 in is equal to 8 ft.
(d) To convert 2,100 yds to mi, we know that 1 mi is equal to 1,760 yds. Setting up the proportion: 1 mi / 1,760 yds = x mi / 2,100 yds. Solving for x, we find x ≈ 1.19 mi. Hence, 2,100 yds is approximately equal to 1.19 mi.
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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
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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.
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For a given function \( f(x) \), the divided-differences table is given by: An approximation of \( f^{\prime}(0) \) is: \( 21 / 2 \) \( 11 / 2 \) \( 1 / 2 \) \( 7 / 2 \)
The approximation of f'(0) using the given divided-differences table is 10.
To approximate f'(0) using the divided-differences table, we can look at the first column of the table, which represents the values of the function evaluated at different points. The divided-differences table is typically used for approximating derivatives by finite differences.
The first column values in the divided-differences table you provided are [tex]\( \frac{21}{2} \), \( \frac{11}{2} \), \( \frac{1}{2} \), and \( \frac{7}{2} \).[/tex]
To approximate f'(0) using the divided-differences table, we can use the formula for the forward difference approximation:
[tex]\[ f'(0) \approx \frac{\Delta f_0}{h}, \][/tex]
where [tex]\( \Delta f_0 \)[/tex] represents the difference between the first two values in the first column of the divided-differences table, and ( h ) is the difference between the corresponding ( x ) values.
In this case, the first two values in the first column are[tex]\( \frac{21}{2} \) and \( \frac{11}{2} \),[/tex] and the corresponding ( x ) values are[tex]\( x_0 = 0 \) and \( x_1 = h \).[/tex] The difference between these values is [tex]\( \Delta f_0 = \frac{21}{2} - \frac{11}{2} = 5 \).[/tex]
The difference between the corresponding ( x ) values can be determined from the given divided-differences table. Looking at the values in the second column, we can see that the difference is [tex]\( h = x_1 - x_0 = \frac{1}{2} \).[/tex]
Substituting these values into the formula, we get:
[tex]\[ f'(0) \approx \frac{\Delta f_0}{h} = \frac{5}{\frac{1}{2}} = 10. \][/tex]
Therefore, the approximation of f'(0) using the given divided-differences table is 10.
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The function f(x) = (x - tan x)/ {x^{3}} has a hole at the point (0, b). Find b.
To find the value of b for the function f(x) = (x - tan(x))/x^3 at the point (0, b), we need to evaluate the limit of the function as x approaches 0. By applying the limit definition, we can determine the value of b.
To find the value of b, we evaluate the limit of the function f(x) as x approaches 0. Taking the limit involves analyzing the behavior of the function as x gets arbitrarily close to 0.
Using the limit definition, we can rewrite the function as f(x) = (x/x^3) - (tan(x)/x^3). As x approaches 0, the first term simplifies to 1/x^2, while the second term approaches 0 because tan(x) approaches 0 as x approaches 0. Therefore, the limit of the function f(x) as x approaches 0 is 1/x^2.
Since we are interested in finding the value of b at the point (0, b), we evaluate the limit of f(x) as x approaches 0. The limit of 1/x^2 as x approaches 0 is ∞. Therefore, the value of b at the point (0, b) is ∞, indicating that there is a hole at the point (0, ∞) on the graph of the function.
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2. Using third order polynomial Interpolation method to plan the following path: A linear axis takes 3 seconds to move from Xo= 15 mm to X-95 mm. (15 Marks)
The third-order polynomial is: f(x) = 15 - 0.00125(x-15)² + 1.3889 x 10^-5(x-15)³
The third-order polynomial interpolation method can be used to plan the path given that the linear axis takes 3 seconds to move from Xo=15 mm to X-95 mm.
The following steps can be taken to plan the path:
Step 1: Write down the data in a table as follows:
X (mm) t (s)15 0.095 1.030 2.065 3.0
Step 2: Calculate the coefficients for the third-order polynomial using the following equation:
f(x) = a0 + a1x + a2x² + a3x³
We can use the following equations to calculate the coefficients:
a0 = f(Xo) = 15
a1 = f'(Xo) = 0
a2 = (3(X-Xo)² - 2(X-Xo)³)/(t²)
a3 = (2(X-Xo)³ - 3(X-Xo)²t)/(t³)
We need to calculate the coefficients for X= -95 mm. So, Xo= 15mm and t= 3s.
Substituting the values, we get:
a0 = 15
a1 = 0
a2 = -0.00125
a3 = 1.3889 x 10^-5
Thus, the third-order polynomial is:f(x) = 15 - 0.00125(x-15)² + 1.3889 x 10^-5(x-15)³
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A steep mountain is inclined 74 degree to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 920 ft out in the plain from the base of the mountain. Find the shortest length of cable needed. Round your answer to the nearest foot.
The shortest length of cable needed is ft
The shortest length ( hypotenuse) of cable needed is approximately 3500 ft (rounded to the nearest foot).
To find the shortest length of cable needed, we can use trigonometry to calculate the hypotenuse of a right triangle formed by the height of the mountain and the horizontal distance from the base of the mountain to the cable car installation point.
Let's break down the given information:
- The mountain is inclined at an angle of 74 degrees to the horizontal.
- The mountain rises to a height of 3400 ft above the surrounding plain.
- The cable car installation point is 920 ft out in the plain from the base of the mountain.
We can use the sine function to relate the angle and the height of the mountain:
sin(angle) = opposite/hypotenuse
In this case, the opposite side is the height of the mountain, and the hypotenuse is the length of the cable car needed. We can rearrange the equation to solve for the hypotenuse:
hypotenuse = opposite/sin(angle)
hypotenuse = 3400 ft / sin(74 degrees)
hypotenuse ≈ 3500.49 ft (rounded to 2 decimal places)
So, the shortest length of cable needed is approximately 3500 ft (rounded to the nearest foot).
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