To evaluate the integral ∫(x^3+4x)/x^4+8x^2+1, we can use the substitution u = x^2 + 1. Then, du/dx = 2x, which means that dx = du/(2x). Substituting these into the integral, we get:
∫(x^3+4x)/x^4+8x^2+1 dx = ∫(1/u)(x^2+1)(x^3+4x)/(2x) du
= 1/2 ∫(u-1)/u^2 du
= 1/2 ∫(u/u^2 - 1/u^2) du
= 1/2 ln|u| + 1/2 (1/u) + C
= 1/2 ln|x^2+1| + 1/2 (1/(x^2+1)) + C
Therefore, the final answer is ∫(x^3+4x)/x^4+8x^2+1 dx = 1/2 ln|x^2+1| + 1/2 (1/(x^2+1)) + C.
Hi! To evaluate the integral, we can rewrite the given expression as follows:
∫((x^3 + 4x) / (x^4 + 8x^2 + 1)) dx
Now, let's use substitution to solve this integral. Let's set:
u = x^2 + 4
Then, the derivative du/dx = 2x. So, dx = du / (2x).
Now, we can rewrite the integral in terms of u:
∫((x^3 + 4x) / (u^2 + 1)) (du / (2x))
Notice that x^3/x and 4x/x simplify, and we are left with:
(1/2) ∫(u / (u^2 + 1)) du
Now we can integrate this expression:
(1/2) * [ln(u^2 + 1) + C]
Now, substitute back x^2 + 4 for u:
(1/2) * [ln(x^2 + 4 + 1) + C] = (1/2) * [ln(x^2 + 5) + C]
So, the evaluated integral is:
(1/2) * [ln(x^2 + 5) + C]
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The temperature at any point (x, y) in a steel plate is T = 900 − 0.7x2 − 1.3y2, where x and y are measured in meters. At the point (3, 9), find the rates of change of the temperature with respect to the distances moved along the plate in the directions of the x- and y-axes.
At point (3, 9), the rate of change of temperature with respect to the x-axis is -4.2 °C/m, and with respect to the y-axis, it is -23.4 °C/m.
To find the rates of change of the temperature with respect to the distances moved along the x- and y-axes at point (3, 9), you need to compute the partial derivatives of the temperature function T(x, y) = 900 - 0.7x^2 - 1.3y^2 with respect to x and y.
For the x-axis:
∂T/∂x = -1.4x
At point (3, 9), ∂T/∂x = -1.4(3) = -4.2 °C/m
For the y-axis:
∂T/∂y = -2.6y
At point (3, 9), ∂T/∂y = -2.6(9) = -23.4 °C/m
So, at point (3, 9), the rate of change of temperature with respect to the x-axis is -4.2 °C/m, and with respect to the y-axis, it is -23.4 °C/m.
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The possible values of 'r' in a-bq + r are 0, 1, 2, 3, 4 and q = 4 then the possible maximum value is
A) 20
B) 25
C) 24
D) None
The possible maximum value of the expression is (d) None
Calculating the possible maximum value of the expressionFrom the question, we have the following parameters that can be used in our computation:
a = bq + r
The above expression is an Euclid's Division statement
The Euclid's Division Algorithm states that "For any two positive integers a, b there exists unique integers q and r such that:"
a = bq + r
where 0 ≤ r < b.
From the question, we have
q = 4
Max r = 4
Using 0 ≤ r < b, we have
Minimum b = 5
So, we have
a = bq + r
This gives
Min a = 5 * 4 + 4
Min a = 24
The above represents the minimum value of a
The maximum value cannot be calculated because as b increases, the value of the expression also increases
Hence, the possible maximum value is (d) None
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The floor of a storage unit is 6.4 feet long and 6.2 feet wide. What is the distance between two opposite corners of the floor? If necessary, round to the nearest tenth.
Answer: 8.9
Step-by-step explanation:
This is pythagoras.
[tex]a^{2} + b^{2} = c^{2}\\6.4^{2} + 6.2^{2} = c^{2}\\40.96 + 38.44 = 79.4\\\sqrt{79.4} = 8.91066776398[/tex]
8.91066776398 = 8.9 (nearest tenth)
!!this is for financial mathematics!! please check if I am correct, thank youu :)
The total interest on a 20-year, 5.26% loan with a principal of $50,000 is $52,600.
How to calculate the interestFrom the information,
Principal: $50,000
Interest rate: 5.26%
Loan duration: 20 years
Total Interest = (Principal x Interest Rate x Loan Duration) / 100
Plugging in the values we have, we get:
Total Interest = (50,000 x 5.26 x 20) / 100
Total Interest = $52,600
The interest is $52600.
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The screen of a 32-inch high definition television has a diagonal length of 31. 5 inches. If the TV screen is 27. 5 inches wide, find the height of screen to the nearest tenth of an inch.
The height of the TV screen is?
Using the Pythagorean theorem we get , the height of the TV screen is approximately 15.4 inches to the nearest tenth of an inch.
The screen of a 32-inch high definition television has a diagonal length of 31.5 inches. If the TV screen is 27.5 inches wide, you need to find the height of the screen to the nearest tenth of an inch. To do this, you can use the Pythagorean theorem, which states that the square of the length of the hypotenuse (diagonal) of a right triangle is equal to the sum of the squares of the other two sides (width and height).
1. Let the height of the TV screen be h inches.
2. According to the Pythagorean theorem, (width)^2 + (height)^2 = (diagonal)^2.
3. Substitute the given values: (27.5)^2 + (h)^2 = (31.5)^2.
4. Calculate the squares: 756.25 + h^2 = 992.25.
5. Subtract 756.25 from both sides: h^2 = 236.
6. Find the square root of 236: h ≈ 15.4 inches.
The height of the TV screen is approximately 15.4 inches to the nearest tenth of an inch.
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find the distance between each pair of points. (5 1/2, -7 1/2) and (5 1/2, -1 1/2)
Answer:
6
Step-by-step explanation:
The distance between both those points are 6
If 3/10 of a number is equal to 1/4 what is the number
Answer:
10/12
Step-by-step explanation:
(3/10)x=1/4
3x=10/4
x=10/12
please show all work so i can understand! thanks!!
Classify each series as absolutely convergent conditionally convergent, or divergent. DO «Σ a (-1)k+1 k! k=1 b. Σ ka sin 2
For series Σ a(-1)^(k+1)k!, convergence depends on the limit of |a(k+1)/a(k)|. For series Σ ka sin(2), it diverges.
Consider the series Σ a(-1)^(k+1)k!, where a is a sequence of real numbers.
To determine the convergence of this series, we can use the ratio test
lim┬(k→∞)〖|a(k+1)(-1)^(k+2)(k+1)!|/|ak(-1)^(k+1)k!| = lim┬(k→∞)〖(k+1)|a(k+1)|/|a(k)||〗
If this limit is less than 1, then the series converges absolutely. If the limit is greater than 1, the series diverges. If the limit is equal to 1, then the test is inconclusive.
Let's evaluate the limit
lim┬(k→∞)〖(k+1)|a(k+1)|/|a(k)||〗 = lim┬(k→∞)〖(k+1)!/(k!k)|a(k+1)/a(k)||〗 = lim┬(k→∞)〖(k+1)/(k)|a(k+1)/a(k)||〗
Since lim┬(k→∞)〖|a(k+1)/a(k)||〗 exists, we can apply the ratio test again:
if the limit is less than 1, the series converges absolutely.
if the limit is greater than 1, the series diverges.
if the limit is equal to 1, the test is inconclusive.
Therefore, we can classify the series Σ a(-1)^(k+1)k! as either absolutely convergent, conditionally convergent, or divergent depending on the value of the limit.
Consider the series Σ ka sin(2), where a is a sequence of real numbers.
To determine the convergence of this series, we can use the alternating series test, which states that if a series Σ (-1)^(k+1)b(k) is alternating and |b(k+1)| <= |b(k)| for all k, and if lim┬(k→∞)〖b(k) = 0〗, then the series converges.
In this case, we have b(k) = ka sin(2), which is alternating since (-1)^(k+1) changes sign for each term. We also have
|b(k+1)|/|b(k)| = (k+1)|a|/k < k|a|/k = |b(k)|/|b(k-1)|
Therefore, |b(k+1)| <= |b(k)| for all k. Finally, we have
lim┬(k→∞)〖b(k) = lim┬(k→∞)〖ka sin(2)〗 = ∞〗
Since the limit does not exist, the series diverges.
Therefore, we can classify the series Σ ka sin(2) as divergent.
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3. Represent and Connect A jar has 20 marbles: 6 black, 4 brown, 8 white, and 2 blue. Julie draw
a marble from the jar.
a. What is the sample space?
b. What is the probability Julie will draw a white marble?
c. Which is more likely to happen, drawing a black marble or drawing either a brown
or blue marble?
d. Using this jar of marbles, what event has a probability of 0?
The event that has a probability of 0 is selecting a yellow marble
Other probabilities are listed below
Identifying the sample space and the probabilitiesThe items in the jar are given as
6 black, 4 brown, 8 white, and 2 blue.
These items are the sample space of this event
Hence, the sample space is 6 black, 4 brown, 8 white, and 2 blue.
For the probability Julie will draw a white marble, we have
P(White) = White/Total
So, we have
P(White) = 8/20
Simplify
P(White) = 2/5
For the event that is more likely to happen, we have
P(black marble) = 6/20
P(brown or blue marble) = (4 + 2)/20
P(brown or blue marble) = 6/20
The probabilities are equal
So, both events have equal likelihood
The event that has a probability of 0 could be the probability of selecting a yellow marble
This is because the jar has no yellow marble
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company A charges $50 per month plus $0.05 per minute. Company B has no monthly charge but charges $0.15 per minute. When is the charge for the companies the same?
The charge for the companies is the same at 500 minutes
When is the charge for the companies the same?From the question, we have the following parameters that can be used in our computation:
Company A charges $50 per month plus $0.05 per minute. Company B has no monthly charge but charges $0.15 per minute.This means that
A(x) = 50 + 0.05x
B(x) = 0.15x
When the charge for the companies are the same, we have
0.15x = 50 + 0.05x
So, we have
0.1x = 50
Divide
x = 500
Hence, the number of minutes is 500
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Point A is located at (−2, 2), and point M is located at (1, 0). If point M is the midpoint of segment AB, find the location of point B.
(−0. 5, 1)
(4, −2)
(−5, 4)
(−1, 1)
The location of point B is (4, −2). So the answer is (4, −2).
How to find the location of the point B?To find the location of point B, we can use the midpoint formula, which states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are:
((x1 + x2)/2, (y1 + y2)/2)
In this case, we know that point M is the midpoint of segment AB, and we know the coordinates of point M. We also know the coordinates of point A. So we can use the midpoint formula to solve for the coordinates of point B.
Let's call the coordinates of point B (x, y). We know that point M is the midpoint of segment AB, so we can set up the following equation:
((−2 + x)/2, (2 + y)/2) = (1, 0)
Simplifying this equation, we get:
(−2 + x)/2 = 1 and (2 + y)/2 = 0
Solving for x and y, we get:
x = 4 and y = −2
Therefore, the location of point B is (4, −2). So the answer is (4, −2).
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helpppp pleaseee!!!!
Answer:
Step-by-step explanation:
During a sale, a store offered a 20% discount on a stereo system that originally sold for $720. After the sale, the discounted price of the stereo system was marked up by 20%. What was the price of the stereo system after the markup? Round to the nearest cent.
The price of the stereo system after the discount and markup is $691.20.
How to determine the markup:The markup price represents the price after adding a percentage of the discounted price.
The markup can be determined using the markup factor, which increases 100% by the markup percentage.
The discount offered on the stereo system = 20%
Original sales price of the system = $720
Discount factor = 0.8 (1 - 0.2)
Discounted price = $576 ($720 x 0.8)
Markup percentage after the discount = 20%
Markup factor = 1.2 (1 + 0.2)
Marked up price = $691.20 ($576 x 1.2)
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if a slope is when xincome in thousands of dollars, then what is the slope when income in dollars? (hint: a one-dollar change has only 1/1000 of the impact of a one-thousand-dollar change.)
If the slope is defined as the change in the dependent variable (y) divided by the change in the independent variable (x), then the slope would depend on how the dependent variable and independent variable are measured.
Assuming that the dependent variable y is some form of economic outcome (e.g., consumption, savings, investment) and the independent variable x is income, then the slope of the relationship would depend on the units in which income is measured.
If income is measured in thousands of dollars, then a one-unit change in income would correspond to a $1,000 change. Therefore, the slope would reflect the change in the economic outcome associated with a $1,000 change in income.
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The value of P from the formula I= PRT/100 when I = 20, R= 5 and T= 4 is ?
The value of P from the formula I= PRT/100 when I = 20, R= 5 and T= 4 is 100.The formula I = PRT/100 is used to calculate the simple interest on a principle amount, where P is the principle amount, R is the interest rate, and T is the time period.
To find the value of P from the formula I = PRT/100 when I = 20, R = 5, and T = 4,
Write down the formula: I = PRT/100 Plug in the given values: 20 = P(5)(4)/100Simplify the equation: 20 = 20P/100 Solve for P: P = 20(100)/20 = 100Therefore the value of P is 100.
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Kadeesha invested $900 in an account that pays 1. 5% interest compounded annually. Assuming no deposits or withdrawals are made, find how much money Kadeesha
would have in the account 11 years after her initial investment. Round to the nearest
tenth (if necessary)
Kadeesha would have approximately $1187.80 in the account 11 years after her initial investment.
We can use the formula for compound interest to solve this problem. The formula is given by:
A = P(1 + r/n)^(nt)
where A is the amount after t years, P is the principal amount, r is the annual interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the number of years.
In this case, P = $900, r = 0.015, n = 1 (since interest is compounded annually), and t = 11. Plugging these values into the formula, we get:
A = $900(1 + 0.015/1)^(1*11) = $1187.7989
Rounding this to the nearest tenth, we get $1187.80. Therefore, Kadeesha would have approximately $1187.80 in the account 11 years after her initial investment.
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Please show me explanations on these
Answer:
1)55,44
2)6,4
3)49,8
4)11,7
5)12,444
6)23,04
7)12,46
8)15,06
9)31,68
10)30,4
Step-by-step explanation:
Recheck them and make sure there are no mistakes
FOIL the equation, don't need to solve!
(2x-1)(x+2)
When we multiply (2x - 1) and (x + 2) using FOIL method, we get:
(2x - 1)(x + 2) = 2x(x) + 2x(2) - 1(x) - 1(2)
= 2x² + 4x - x - 2
= 2x² + 3x - 2
Therefore, the product of (2x - 1) and (x + 2) is 2x² + 3x - 2.
The preimage and image of WXYZ are shown. Use coordinate notation to describe the translation.
The image of WXYZ moves two unit to the right and 5 unit down. The coordinate is (x+2, y-5).
The polar system of coordinates is a coordinate system with two dimensions in which the location of each point can be determined by its distance from the origin, a fixed point, and its angle from the polar axis, also known as the polar axis of rotation. A ray beginning at the origin and extending outward at a set angle, typically 0 degrees or horizontal, is how the polar axis is typically depicted. The image of WXYZ moves two unit to the right and 5 unit down. The coordinate is (x+2, y-5).
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The diameter of Circle Q terminates on the circumference of the circle at (0,3)and (0,−4). Write the equation of the circle in standard form. Show all of your work.
Need answer ASAP Please!!
Answer:
[tex]x^2+\left(y+\dfrac{1}{2}\right)^2=\dfrac{49}{4}[/tex]
Step-by-step explanation:
The center of the circle is the midpoint of its diameter.
[tex]\boxed{\begin{minipage}{7.4 cm}\underline{Midpoint between two points}\\\\Midpoint $=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)$\\\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are the endpoints.\\\end{minipage}}[/tex]
Given the endpoints of the diameter are (0, 3) and (0, -4), to find the coordinates of the center of the circle, substitute the two endpoints into the midpoint formula:
[tex]\text{Center}=\left(\dfrac{0+0}{2},\dfrac{-4+3}{2}\right)=\left(0,-\dfrac{1}{2}\right)[/tex]
As the x-values of the endpoints of the diameter are the same, the length of the diameter, d, is the absolute value of the difference in y-values of the endpoints:
[tex]d=|3-(-4)|=7[/tex]
Therefore, the diameter of circle Q is 7 units.
The radius, r, of a circle is half its diameter. Therefore:
[tex]r=\dfrac{d}{2}=\dfrac{7}{2}[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h,k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]
Now we have determined the center and radius of circle Q, we can substitute these values into the equation of a circle to write the equation of circle Q in standard form:
[tex](x-0)^2+\left(y-\left(-\dfrac{1}{2}\right)\right)^2=\left(\dfrac{7}{2}\right)^2[/tex]
[tex]x^2+\left(y+\dfrac{1}{2}\right)^2=\dfrac{49}{4}[/tex]
Therefore, the equation of circle Q in standard form is:
[tex]\boxed{x^2+\left(y+\dfrac{1}{2}\right)^2=\dfrac{49}{4}}[/tex]
Leland wants to paint a portrait for his parents using a photograph of them that is 5 inches wide by 8 inches high. He wants the portrait to be proportional to the photograph and 42 inches high. Which proportion can Leland use to find , the width he needs to use to make the portrait proportional?
The ratio to make the portrait proportional is width/42 = 5/8
Which proportion can Leland use to find , the width he needs to useFrom the question, we have the following parameters that can be used in our computation:
Photograph that is 5 inches wide by 8 inches high.
This means that
Ratio = width/height
Substitute the known values in the above equation, so, we have the following representation
Ratio = 5/8
Given that he wants the portrait to be proportional to the photograph and 42 inches high.
Then, we have
width/42 = 5/8
Hence, the ratio to make the portrait proportional is width/42 = 5/8
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The figure 2 is dilated from figure 1. Find the scale factor.
There is 1. 75 liter of water in a rectangular container. The base of the container is square on the side 12 cm and its height is 16. 5 cm. How much more water is needed to fill the container to its brim? Give your answer in liter
0.626 liters of water is needed to fill the container to its brim.
The volume of the rectangular container can be found by multiplying the area of the base (length x width) by the height:
Volume of rectangular container = length x width x height
Since the base is a square with a side of 12 cm, the area of the base is:
Area of base = 12 cm x 12 cm = 144 cm^2
Converting the height to cm, we have:
Height = 16.5 cm
So the volume of the container is:
Volume = 144 cm^2 x 16.5 cm = 2376 cm^3
To convert the volume from cubic centimeters to liters, we divide by 1000:
Volume = 2376 cm^3 ÷ 1000 = 2.376 liters
Since there is already 1.75 liters of water in the container, the amount of water needed to fill the container to its brim is:
Amount of water needed = 2.376 liters - 1.75 liters = 0.626 liters
Therefore, 0.626 liters of water is needed to fill the container to its brim.
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Elizabeth is considering buying a $30,000 car. Which of these financing options
will likely lead to the LOWEST monthly payment?
$3000 down payment, 6% interest, 84 months
$3000 down payment, 6% interest, 60 months
$0 down payment, 6% interest, 60 months
$0 down payment, 0% interest, 36 months
The financing option that will likely lead to the lowest monthly payment is:
$3000 down payment, 6% interest, 84 months
The longer loan term (84 months) will spread out the payments over a longer period of time, resulting in a lower monthly payment. The down payment will also help to reduce the monthly payment amount.
The 6% interest rate is relatively low, so it won't have a significant impact on the monthly payment compared to the loan term.
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if the atlanta hawks free throw percentage is 82%, what is the probability that a player for the hawks will make 2 free shots in a row?
Answer:
Approx 67.24%
Step-by-step explanation: If the Atlanta Hawks' free-throw percentage is 82%, the probability that a player will make one free throw is 0.82.
To find the probability that a player will make two free throws in a row, we can use the multiplication rule of probability which states that the probability of two independent events occurring together is the product of their individual probabilities.
Therefore, the probability of a player making two free throws in a row can be calculated as follows:
P(making two free throws in a row) = P(making first free throw) x P(making second free throw)
P(making two free throws in a row) = 0.82 x 0.82
P(making two free throws in a row) = 0.6724 or 67.24%
Therefore, the probability that a player for the Atlanta Hawks will make two free shots in a row is approximately 67.24%
Two forces of 39n (newtons) and 46n act on an object at right angles. find the magnitude of the resultant and the angle that it makes with the smaller force.
The magnitude of the resultant force is approximately 60.28 newtons. The angle between the resultant force and the smaller force is approximately 50.5 degrees.
To find the magnitude of the resultant force, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, the two forces are acting at right angles, so we can treat them as the sides of a right triangle:
resultant force^2 = (39n)^2 + (46n)^2
resultant force^2 = 1521n^2 + 2116n^2
resultant force^2 = 3637n^2
resultant force = sqrt(3637n^2) = 60.28n
So the magnitude of the resultant force is approximately 60.28 newtons.
To find the angle that the resultant force makes with the smaller force, we can use trigonometry.
We know that the two forces are at right angles, so the angle between the resultant force and the smaller force is the same as the angle between the resultant force and the larger force. Let's call this angle θ. Then we have:
tan θ = (larger force) / (smaller force)
tan θ = 46n / 39n
θ = tan^-1(46/39) = 50.5°
Therefore, the angle between the resultant force and the smaller force is approximately 50.5 degrees.
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The likelihood that a child will attend a live musical performance can be modeled by
q = 0.01(0.0005x2 + 0.38x + 35) (15 ≤ x ≤ 100).
Here, q is the fraction of children with annual household income x thousand dollars who will attend a live musical performance during the year. Compute the income elasticity of demand E at an income level of $30,000. (Round your answer to two significant digits.)
E =
Interpret the result.
At a family income level of $_______ , the fraction of children attending a live musical performance is increasing by ________% per 1% increase in household income.
E = 0.01(0.0005(30)^2 + 0.38(30) + 35)(2*0.0005(30) + 0.38) ≈ 0.63
Interpretation: The income elasticity of demand is 0.63, which means that for every 1% increase in household income, the fraction of children attending a live musical performance increases by 0.63%.
At a family income level of $30,000, the fraction of children attending a live musical performance is increasing by 0.63% per 1% increase in household income.
To compute the income elasticity of demand E at an income level of $30,000, we need to find the derivative of q with respect to x, and then evaluate it at x=30. The derivative of q with respect to x is:
dq/dx = 0.01(0.001x + 0.38)
Now, let's evaluate this derivative at x=30:
dq/dx = 0.01(0.001(30) + 0.38) = 0.004
To calculate the income elasticity of demand E, we use the formula:
E = (dq/dx)(x/q)
First, let's find q at x=30:
q = 0.01(0.0005(30)^2 + 0.38(30) + 35) = 0.178
Now, we can find E:
E = (0.004)(30/0.178) ≈ 0.67
Interpret the result:
At a family income level of $30,000, the fraction of children attending a live musical performance is increasing by approximately 67% per 1% increase in household income.
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The equation of the line of best fit relating age (in years) and the median height (in cm) of boys is given.
the slope of [tex]6.5[/tex] in this equation indicates that for each additional year of age, the median height of boys increases by approximately 6.5 cm. Thus, option D is correct.
What is median?The statement that best interprets the slope in the context of the problem is "The slope is 6.5, this means that each year boys grow approximately [tex]6.5[/tex] cm."
The slope of a linear equation represents the rate of change, or the amount by which the dependent variable (in this case, median height) changes for each unit increase in the independent variable (in this case, age).
Therefore, the slope of [tex]6.5[/tex] in this equation indicates that for each additional year of age, the median height of boys increases by approximately [tex]6.5[/tex] cm.
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Which of the following is an even function? g(x) = (x – 1)2 1 g(x) = 2x2 1 g(x) = 4x 2 g(x) = 2x
The only even function among the given options is g(x) = 2x^2, so the answer is B) g(x) = 2x^2.
A function is even if it satisfies the property g(-x) = g(x) for all x.
Checking each of the given functions:
g(x) = (x - 1)^2 is not even, because g(-x) = (-x - 1)^2 = x^2 + 2x + 1, which is not equal to g(x) = (x - 1)^2.
g(x) = 2x^2 is even, because g(-x) = 2(-x)^2 = 2x^2 = g(x) for all x.
g(x) = 4x^2 is even, because g(-x) = 4(-x)^2 = 4x^2 = g(x) for all x.
g(x) = 2x is odd, because g(-x) = 2(-x) = -2x = -g(x) for all x.
Therefore, the only even function among the given options is g(x) = 2x^2, so the answer is B) g(x) = 2x^2.
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Consider the function f(x) = x^3 – 5x on the closed interval [ - 1,3). Find the exact value of the slope of the secant line connecting ( - 1, f( - 1)) and (3, f(3)). m = ?
To find the slope of the secant line connecting two points on a curve, we can use the formula: slope of secant line = (change in y) / (change in x).
In this problem, we are given the function f(x) = x^3 - 5x on the closed interval [-1, 3), and we need to find the slope of the secant line connecting (-1, f(-1)) and (3, f(3)). To do this, we first need to find the y-coordinates of these points by plugging the given x-values into the function f(x). Then, we can use the formula for the slope of a secant line to find the slope of the line connecting these two points.
The slope of a secant line is an important concept in calculus and is used to approximate the instantaneous rate of change of a function at a particular point.
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