That since the rectangle is a square, both values are the same.
Let the length of the rectangle be L and its width be W.
The area of the rectangle is given by A = LW, and the perimeter is given by P = 2L + 2W.
We want to minimize the perimeter subject to the constraint that the area is 1000 m^2.
From the area equation, we can solve for L in terms of W: L = 1000/W.
Substituting this expression for L into the perimeter equation, we get:
P = 2(1000/W) + 2W = 2000/W + 2W
To find the minimum value of P, we take the derivative of P with respect to W and set it equal to zero:
dP/dW = -2000/W^2 + 2 = 0
Solving for W, we get:
W = sqrt(1000) = 31.62 m
Substituting this value for W into the equation for L, we get:
L = 1000/W = 1000/31.62 = 31.62 m
Therefore, the dimensions of the rectangle with area 1000 m^2 and minimum perimeter are:
Length = 31.62 m
Width = 31.62 m
Note that since the rectangle is a square, both values are the same.
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10. What is the radius of a sphere with a volume of 4186 in³ to the nearest tenth of an inch?
Answer:10
Step-by-step explanation:
10
Find the surface area of the regular pyramid.
The surface area of the pyramid is 141 in²
How to find the area of the regular pyramid?Considering the figure, to find the surface area of the regular pyramid, we notice that 3 similar triangular faces and 1 other triangular face.
So, the surface area of the pyramid is A = 3A' + A where A' = 1/2bH where
b = 6 in and H = 14 inSo, A' = 1/2bh
= 1/2 × 6 in × 14 in
= 3 in × 14 in
= 42 in²
Also,
A' = 1/2bh where
b = 6 in and h = 5.2 inSo, A' = 1/2bh
= 1/2 × 6 in × 5 in
= 3 in × 5 in
= 15 in²
So, A = 3A' + A"
= 3 × 42 in² + 15 in²
= 126 in² + 15 in²
= 141 in²
So, the surface area of the pyramid is 141 in²
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Andrew and Ruth Bacon would like to obtain an installment loan of 1850 to repaint their home. They can get the loan at an APR of a) 8% for 24 months or b) 11% for 18 months. Which loan has the lower finance charge?
8% for 24 months has the lower finance charge of $296, compared to option b) with a finance charge of $363.50.
To compare the two loans, we need to calculate the finance charge for each option.
For option a) at 8% APR for 24 months, we can use the following formula:
Finance charge = (loan amount x interest rate x time) / 12
Finance charge = (1850 x 0.08 x 24) / 12 = 296
So the finance charge for option a) is $296.
For option b) at 11% APR for 18 months, we can use the same formula:
Finance charge = (loan amount x interest rate x time) / 12
Finance charge = (1850 x 0.11 x 18) / 12 = 363.5
So the finance charge for option b) is $363.50.
Therefore, option a) has the lower finance charge of $296, compared to option b) with a finance charge of $363.50.
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State the appropriate test statistic name, degrees of freedom, test statistic value, and the associated p-value (Enter your degrees of freedom as a whole number, the test statistic value to three decimal places, and the p-value to four decimal places).t(45) = ________ p= ________
Degrees of freedom (df) refers to the number of independent pieces of information that can be used to estimate a parameter. The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated from your sample data, assuming the null hypothesis is true.
However, I can still help you understand the terms and how they relate to your question.
1. Test Statistic Name: In this case, the test statistic is the t-statistic, which is used for hypothesis testing in statistics when the population standard deviation is unknown.
2. Degrees of Freedom: Degrees of freedom (df) refers to the number of independent pieces of information that can be used to estimate a parameter. In a t-test, the degrees of freedom are typically represented as "t(df)". In your example, the degrees of freedom are 45 (t(45)).
3. Test Statistic Value: This is the calculated value of the t-statistic, which you will need to compute based on the data provided. It is used to compare against the critical value or to find the p-value. You need to provide the data or information about the test to calculate this value.
4. P-value: The p-value is the probability of observing a test statistic as extreme or more extreme than the one calculated from your sample data, assuming the null hypothesis is true. You will need to compute the p-value using the t-statistic value.
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The alternating series test can be used to show convergence of which of the following alternating series?I. 4−19+1−181+14−1729+116−...,+an+...,where an={82nif n is odd−13nif n is evenII. 1−12+13−14+15−16+17−18+...+an+...,where an(−1)n+1nIII. 23−35+47−59+611−713+815−...+an+...,where an=(−1)n+1n+12n+1(A) I only(B) II only(C) III only(D) I and II only(E) I, II, and III
The alternating series test can be used to show convergence of the alternating series I, II, and III given in the options and the correct answer to this question is Option A. I only.
The alternating series test is a method used to determine the convergence or divergence of alternating series. According to the alternating series test, an alternating series converges if the absolute value of its terms decreases monotonically to zero. In other words, if the absolute value of the terms in an alternating series eventually becomes smaller and smaller until it is less than or equal to a certain positive number, then the series converges.
In series, I, the absolute value of the terms decreases monotonically to zero since the terms eventually become smaller and smaller. Therefore, series I converge by the alternating series test.In series II, the absolute value of the terms does not decrease monotonically to zero, since the terms eventually increase in magnitude. Therefore, the alternating series test cannot be used to show the convergence or divergence of series II.In series III, the absolute value of the terms decreases monotonically to zero since the terms eventually become smaller and smaller. Therefore, series III converges by the alternating series test.In conclusion, the alternating series test can be used to show the convergence of series I and III, but not for series II. Therefore, the answer is (A) I only.
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What is the value of the expression −3 1/3÷(−2.4) ?
Answer:
First, we need to convert the mixed number −3 1/3 to a fraction. −3 1/3 = −(3 + 1/3) = −(10/3).
Now, we can divide the fraction by the decimal. −(10/3) ÷ (-2.4) = −(10/3) ÷ (-24/10) = −(10/3) x (10/-24) = 10/-7.2 = -1.388888889.
Therefore, the value of the expression is −1.388888889.
Step-by-step explanation:
1. Convert the mixed number to a fraction.
```
-3 1/3 = -(3 + 1/3) = -(10/3)
```
2. Multiply the numerator and denominator of the fraction by -1.
```
-(10/3) = (-1)(10/3) = -10/3
```
3. Divide the numerator and denominator of the fraction by -24.
```
-10/3 = (-10/3) ÷ (-24/10) = 10/-7.2 = -1.388888889
```
Therefore, the value of the expression is −1.388888889.
Here is a visual representation of the steps:
```
-3 1/3 ÷ (-2.4)
= -(10/3) ÷ (-24/10)
= -(10/3) x (10/-24)
= 10/-7.2
= -1.388888889
```
14 1 point If two parents are homozygous for a genetically inherited recessive trait, what is the probability that they will have a child who does not have this trait in his or her phenotype?
The child will always have the recessive trait in their phenotype.
If both parents are homozygous for a recessive trait, it means they both carry two copies of the recessive allele. Let's assume that the dominant allele is represented by 'A' and the recessive allele by 'a'. Since both parents are homozygous for the recessive trait, their genotype must be 'aa'.
When these parents have children, they will each contribute one 'a' allele, resulting in all of their children inheriting the recessive allele. The probability that their child will have the trait is therefore 100%. The probability of not inheriting the trait is 0%.
Therefore, the answer to the question is 0%. The child will always have the recessive trait in their phenotype.
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Wingate Metal Products, Inc. sells materials to contractors who construct metal warehouses, storage buildings, and other structures. The firm has estimated its weighted average cost of capital to be 9.0 percent based on the fact that its after-tax cost of debt financing was 7 percent and its cost of equity was 12 percent.
What are the firm's capital structure weights (that is, the proportions of financing that came from debt and equity)?
Wingate Metal Products, Inc.'s capital structure weights are 60% for debt financing and 40% for equity financing
To find Wingate Metal Products, Inc.'s capital structure weights for debt and equity financing, you need to first identify the weighted average cost of capital (WACC), after-tax cost of debt financing, and cost of equity financing.
The information provided is as follows:
- WACC: 9.0%
- After-tax cost of debt financing: 7%
- Cost of equity financing: 12%
Let's use the formula for WACC:
WACC = (Weight of Debt * Cost of Debt) + (Weight of Equity * Cost of Equity)
Since the weights of debt and equity financing must sum up to 1, we can represent the weight of debt as "x" and the weight of equity as "1-x". Now, we can rewrite the formula:
9.0% = (x * 7%) + ((1-x) * 12%)
Now, solve for x (weight of debt financing) and 1-x (weight of equity financing):
9.0% = 7x + 12 - 12x
9.0% = 12 - 5x
5x = 3%
x = 0.6
The weight of debt financing is 0.6, and the weight of equity financing is 1-0.6 = 0.4.
Therefore, Wingate Metal Products, Inc.'s capital structure weights are 60% for debt financing and 40% for equity financing.
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The base of this right triangular prism is a right triangle with legs that are 7 in. and 8 in. The height of the prism is 5 in.
What is the volume of this right triangular prism?
plsss help
Step-by-step explanation:
Area of base ( 1/2 * L1 * L2 ) * height = volume
1/2 ( 7)(8) * 5 = 140 in^3
on a roulette table, you can bet on a single number (a single square, corresponding to a single slot on the wheel). there are 38 numbers, so the odds are 37 to 1 against you. if you risk $1 on a single square and win, you get it back plus $35 in winnings. (a) if you bet on a single number for 25 rounds, what is the expected value of your net gain? (b) the standard deviation of your net gain? (c) estimate the chance you come out ahead.
(a) If you bet on a single number for 25 rounds, the expected value of your net gain is -$1.32.
(b) The standard deviation of your net gain is 28.0126.
(c) The estimate the chance you come out ahead is 48.17%.
(a) The probability of winning a single bet is 1/38 and the expected net gain for each bet is -1 + (35/1)1/38 = -0.0526. Therefore, the expected value of your net gain after 25 rounds is 25(-0.0526) = -$1.32.
(b) The variance of the net gain for each bet is [(-1 - (-0.0526))^2*(37/38) + (35 - (-0.0526))^2*(1/38)] = 31.3728. So, the variance of the net gain for 25 rounds is 25*31.3728 = 784.32, and the standard deviation is the square root of the variance, which is 28.0126.
(c) The chance of coming out ahead can be estimated using the normal distribution with mean -1.32 and standard deviation 28.0126. We want to find the probability that the net gain is greater than zero, which is equivalent to finding the probability that a standard normal random variable Z is greater than (0 - (-1.32))/28.0126 = 0.0471.
Using a standard normal table or calculator, we find that this probability is approximately 0.4817 or 48.17%. So, there is about a 48.17% chance of coming out ahead after 25 rounds of betting on a single number.
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What is the slope of a line passing through the points (5,4) and (10,14)
1. Find any extrema or saddle points of f(x,y) = x^3 + 12xy - 3y^2 - 27x + 34 2. A company plans to manufacture closed rectangular boxes that have a volume of 16 ft? Without using Lagrange multipliers, find the dimensions that will minimize the cost if the material for the top and bottom costs twice as much as the material for the sides
The dimensions that minimize the cost subject to the volume constraint are [tex]L = 4 ft, W = 2 ft,[/tex] and [tex]H = 2 ft[/tex] using surface area.
Assuming that the cost of material is proportional to the surface area, we can write the cost function as:
[tex]C = k(2LW + 2LH + WH)[/tex]
where k is a constant of proportionality that depends on the cost of the material. We are given that the cost of the material for the top and bottom is twice the cost of the material for the sides, so we can take k = 3 for simplicity (since the cost of the material for the sides is then 1).
Using the volume constraint as before, we can eliminate one of the variables:
[tex]H = 16/LW[/tex]
When this is used as a cost function substitution,
[tex]C = 3(2LW + 2LH + WH) = 6LW + 96/L + 48/W[/tex]
To find the critical points of C, we need to find where the partial derivatives are zero:
[tex]dC/dL = 6W - 96/L^2 = 0[/tex]
[tex]dC/dW = 6L - 48/W^2 = 0[/tex]
When we simultaneously solve these equations, we obtain:
L = 4 ft
W = 2 ft
H = 2 ft
Therefore, the dimensions that minimize the cost subject to the volume constraint surface area are L = 4 ft, W = 2 ft, and H = 2 ft.
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#4- Find the volume of the right prism. Round your answer to two decimal places, if necessary.
Thank you
I’m a bit confused. I know the formula is V=Bh
The base is the 2 rectangles on the side right? I just can’t find the height.
To find the volume of the right prism, we used the Pythagorean theorem to determine the height of the triangular base is 1.197 inches. We then used the formula V = Bh to calculate the volume, which was approximately 2.70 cubic inches.
To find the height of the prism, we need to use the information provided about the triangular base. Since the triangular base is equilateral with a dimension of 1.74 inches, the height of the triangle (and therefore, the height of the prism) can be found by using the Pythagorean theorem.
If we draw a line from the center of the base to the midpoint of one of the sides, we create a right triangle with hypotenuse 1.74 in (which is also the height of the triangle) and one leg equal to half the length of one of the sides of the triangle (since the base of the prism is a square with dimension 1.5 in).
Using the Pythagorean theorem, we can solve for the height of the triangle (and prism)
(1.74/2)² + (1.5/2)² = h²
0.8725 + 0.5625 = h²
h² = 1.435
h ≈ 1.197 inches
Now, we can use the formula V = Bh to find the volume of the prism
V = (1.5 x 1.5) x 1.197 ≈ 2.70 cubic inches
Therefore, the volume of the right prism is approximately 2.70 cubic inches.
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need extreme help with my math
Given that a ski set is being sold at 10% discount at $325, we need to find its original price,
Let the original price be x,
Therefore,
90% of x = 325
0.9x = 325
x = 361.11
Hence the original price of the ski set is $361.11.
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Suppose two females are randomly selected. What is the probability both survived
The probability that both females survived is 0.2961
What is the probability both survivedThe table of values is given as
Male Female Child Total
survived 230 339 54 623
died 1190 102 52 1344
total 1420 441 106 1967
For females that survived, we have
P(Female) = 339/623
For two females, we have
P = 339/623 * 339/623
Evaluate
P = 0.2961
Hence, the probability is 0.2961
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AABC is rotated 90° clockwise about the origin.
-48
-16
-14
12
10
4
N
C(6. 12)
B(18,6)
A(12, 0)
24 6 8 10 12 14 16 18
What are the coordinates of B'?
A. (-18,6)
B. (-6, 18)
C. (6, 18)
D. (6, -18)
Can you please help me with these three problems? I’m really confused about this unit.
The angles are 11°, 42° and 35°.
Given are circles, we need to find the missing angles,
1) ∠1 = 1/2 [119° - (360° - (119°+174°)]
= 1/2 [119° - 97°]
∠1 = 11°
2) ∠1 = 1/2[360°-138°-138°]
∠1 = 1/2 x 84
∠1 = 42°
3) ∠1 = 1/2[111°-360°-(111°+104°+104°)]
∠1 = 1/2 x 70
∠1 = 35°
Hence the angles are 11°, 42° and 35°.
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A fenced backyard has a length
of 20 feet, and width of 25 feet,
and a diagonal of 30 feet. Does
the backyard have a 90 degree
angle in its corner?
Answer:no it doesn’t it makes a trapezoid which doesn’t have 90 degree angles or right angles
Step-by-step explanation:
How much rain would have to fall in August so that the total amount of rain equals the average rainfall for these three months? What would the departure from the average be in August in that situation?
0.86 inches. The departure from the average would then be 0.86 inches since 2.43-1.57=0.86 inches.
a) 0.5 inches. The difference between the average rainfall and the actual rainfall for last June is 0.67-0.17=0.50 inches.
b) 1.14 inches. Because the departure from the average was negative, the actual rainfall was 0.36 inches less than the average rainfall. Thus, 1.5-0.36=1.14 inches was the actual rainfall last July.
c) 2.43 inches. The average rainfall for these three months is 0.67+1.5+1.57=3.74 inches. Last June it rained 0.17 inches and last July it rained 1.14 inches. So, it would need to have rained 3.74-0.17-1.14=2.43 inches last August so that the total amount of rain equaled the average rainfall for these three months.
0.86 inches. The departure from the average would then be 0.86 inches since 2.43-1.57=0.86 inches.
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Full Question ;
The departure from the average is the difference between the actual amount of rain and the average amount of rain for a given month.
The historical average for rainfall in Albuquerque, NM for June, July, and August is shown in the table.
June 0.67 inches
July 1.5 inches
August 1.57 inches
Last June only 0.17 inches of rain fell all month. What is the difference between the average rainfall and the actual rainfall for last June? Answer with decimals.
The departure from the average rainfall last July was -0.36 inches. How much rain fell last July? Answer with decimals.
How much rain would have to fall in August so that the total amount of rain equals the average rainfall for these three months? Answer with decimals.
What would the departure from the average be in August in that situation? Answer with decimals.
calculate the slope of the lines that pass through 5,-8 and -3,-4
Slope of the lines that pass through points (5,-8) and (-3,-4) is -1/2
The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is
m=y₂-y₁/x₂-x₁
We have to find slope of the lines that pass through (5,-8) and (-3,-4)
Slope = -4-(-8)/-3-5
=-4+8/-8
=-4/8
=-1/2
Hence, slope of the lines that pass through (5,-8) and (-3,-4) is -1/2
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when a fuction is divided by 2x-5 and the quotient is 2x^2-2x-3 and the remainder is -8, find the function and write in standard form
To find the function when it is divided by 2x-5 with a quotient of 2x^2-2x-3 and a remainder of -8, follow these steps:
1. Set up the division equation: function = (divisor × quotient) + remainder
2. Substitute the given terms: function = ((2x-5) × (2x^2-2x-3)) - 8
Now, expand and simplify the equation:
3. Multiply the divisor and quotient: function = (4x^3 - 4x^2 - 6x - 10x^2 + 10x + 15) - 8
4. Combine like terms: function = 4x^3 - 14x^2 + 4x + 7
The function in standard form is 4x^3 - 14x^2 + 4x + 7.
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I need help fast please
The probability that the person chosen belonged to Group Y is 69/164.
As, Out of 200 persons in the sample, those having at least one dream are 200− those who had no dream are
= 200−36
=164
Now, out of 164 people belonged to group Y
= 100−21
=79
So, the probability that the person chosen belonged to Group Y become
= 69/164
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Solve the separable differential equation dy / dx = − 0. 6 y , and find the particular solution satisfying the initial condition y (0) = − 9. Y(x) =___
The differential equation solution is y(x) = [tex]-9e^_{(-0.6x)[/tex] for the initial condition y(0) = -9.
The given differential condition is dy/dx = -0.6y. To address this condition, we can isolate the factors by partitioning the two sides by y and duplicating the two sides by dx:
dy/y = -0.6dx
Then, we can incorporate the two sides. On the left side, we get ln|y|, and on the right side, we get -0.6x+C, where C is an inconsistent steady of joining:
ln|y| = -0.6x+C
To find the specific arrangement that fulfills the underlying condition y(0) = -9, we can substitute x = 0 and y = -9 into the situation:
ln|-9| = -0.6(0)+C
Rearranging, we get:
C = ln(9)
In this manner, the specific arrangement that fulfills the underlying condition is:
y(x) = [tex]-9e^_{(-0.6x)[/tex]
This is the last response.
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The complete question is:
Solve the separable differential equation dy/dx = 0.9y, and find the particular solution satisfying the initial condition y(0) = -9. y(x) = e^((0.9/2)x^2-ln(9)).
For the IVP: (t-4) cos ty" – In(t-1)y'+√7+5y=e-', y(2) = 1, y'(2) = 1 determine the largest interval in which the solution is certain to exist
a. (-5,4)
b. (π/2,4)
c. (1,[infinity])
d. (1,π/2)
We can conclude that the largest interval in which the solution is certain to exist is (1,π/2).
To determine the largest interval in which the solution is certain to exist, we need to check the coefficients and initial values for any discontinuities or singularities.
Notice that the coefficient of the second derivative term, (t-4)cos(ty''), becomes zero at t=4, which can cause a singularity in the solution. Moreover, the coefficient of the first derivative term, In(t-1), becomes negative for t<1, which can cause instability issues in the solution.
Since the initial value problem is given for t=2, the interval of certain existence must contain t=2. Therefore, we can eliminate option a (-5,4) and option b (π/2,4) since neither of them contain t=2.
For option c (1,[infinity]), the coefficient of the first derivative term becomes negative for t<1, which violates the condition for the existence of a solution. Therefore, option c can also be eliminated.
The only remaining option is d (1,π/2). This interval contains t=2 and does not cause any discontinuity or instability issues in the coefficients. Therefore, we can conclude that the largest interval in which the solution is certain to exist is (1,π/2).
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Pls Reply before tommorrow
1. A bathtub is being filled at a rate of 2.5 gallons per minute. The bathtub will
hold 20 gallons of water.
a. How long will it take to fill the bathtub?
b. Is the relationship described linear, inverse, exponential, or neither? Write
an equation relating the variables.
2. Suppose a single bacterium lands on one of your teeth and starts reproducing
by a factor of 4 every hour.
a. After how many hours will there be at least 1,000,000 bacteria in the new
colony?
b. Is the relationship described linear, inverse, exponential, or neither? Write
an equation relating the variables.
1.
a.
It will take 8 minutes to fill the bathtub.
b.
The relationship described is linear.
The equation is 20 = 2.5t + 0.
2.
a.
It will take approximately 4.807 hours to have at least 1,000,000 bacteria in the new colony.
b.
The relationship described is exponential,
The equation is Number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
We have,
1.
a.
To fill the bathtub, we need 20 gallons of water.
The rate at which the water is being filled is 2.5 gallons per minute.
Using the formula:
time = amount/rate
we get:
time = 20/2.5 = 8 minutes
b.
The relationship described is linear.
The equation relating the variables can be written as:
amount of water = rate x time + initial amount
where the rate is 2.5 gallons per minute, the initial amount is 0 gallons, and the amount of water is 20 gallons.
So, the equation is:
20 = 2.5t + 0
where t is the time in minutes.
2.
a.
The relationship described is exponential.
The equation relating the variables can be written as:
number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
where the initial number of bacteria is 1, the reproduction factor is 4, and we need to find the time it takes to reach 1,000,000 bacteria.
So, we have:
1,000,000 = 1 x 4^(time/hour)
Taking the logarithm of both sides, we get:
log(1,000,000) = log(4^(time/hour))
6 = (time/hour) x log(4)
time/hour = 6/log(4)
time = (6/log(4)) x hour
time ≈ 4.807 hours
b.
The relationship described is exponential, and the equation relating the variables is:
Number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
where the initial number of bacteria is 1, the reproduction factor is 4, and t is the time in hours.
Thus,
1.
a.
It will take 8 minutes to fill the bathtub.
b.
The relationship described is linear.
The equation is 20 = 2.5t + 0.
2.
a.
It will take approximately 4.807 hours to have at least 1,000,000 bacteria in the new colony.
b.
The relationship described is exponential,
The equation is Number of bacteria = initial number of bacteria x (reproduction factor)^(time/hour)
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If 5 liters of a solution are 20% acid, how much of the solution is acid?
0. 2 liters
1 liter
2 liters
Answer:
0.2
Step-by-step explanation:
About 58,000 people live in a circular region with a 2 mile radius. Find the population density in people per square mile.
I NEED HELP QUICK!!!!!!
Answer:
58,000 / (π(2^2))
= 58,000/(4π)
= 14,500/π
= 4,615 people per square mile
y=1/3(7/4)x growth or decay
The equation represents a growth function.
We have,
The equation y = (7/4)x/3 can be simplified to y = (7/12)x.
Since the coefficient of x, 7/12, is positive, this means that as x increases, y also increases.
In other words, y is growing as x increases, and the growth rate is determined by the slope of the line, which is 7/12.
To understand this intuitively, we can think of the equation as representing a line on a graph.
The slope of the line, which is equal to the coefficient of x, tells us whether the line is increasing or decreasing.
In this case, the positive slope tells us that the line is increasing, which means that y is also increasing as x increases.
This is consistent with a growth function.
Therefore,
The equation represents a growth function.
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Need help asap I don’t understand at all please nd thanks
The two points that a line of best fit would go through would be B. (3, 5) and ( 5, 6 ).
Why would a line of best fit go through these ?The line of best fit is determined by the data points that have the least sum of squared distances from the line. These chosen points provide a clear representation of the general trend of the data and effectively facilitate precise future predictions concerning upcoming data points.
The line of best fit would therefore go through (3, 5) and ( 5, 6 ) because it would lead to four points being above the line, and four points being below which would be a good line of best fit.
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The density function of the random variable X is:
-
p(x) =
=
0,
x <1;
1
(x - 1), 1
12
1
3< x < 6;
6
5 1
x, 6< x <10;
12 24
0,
x >10
1
X
a)Make a drawing showing the value of the function depending on the detection area.
b)Write down the corresponding calculation formula and find the average value. (Convert conversions and calculations in detail.)
The expected value of X is 8.
a) Here is a sketch of the density function p(x) with respect to the detection area:
|
|
|
|
|
|
|
|
|
_____________|_____________
1 1.5 3 6 10
b) The formula for the expected value (or mean) of a continuous random variable X with density function p(x) is:
E(X) = ∫xp(x)dx
To find the expected value of X for the given density function, we need to split the integral into several parts based on the different intervals where p(x) takes different forms:
E(X) = ∫_(-∞)^1 xp(x)dx + ∫_1^2 xp(x)dx + ∫_2^3 xp(x)dx + ∫_3^6 xp(x)dx + ∫_6^10 xp(x)dx + ∫_10^∞ xp(x)dx
Note that the first and last integrals are both zero, since p(x) = 0 for x < 1 and x > 10. The other integrals can be evaluated as follows:
∫_1^2 xp(x)dx = ∫_1^2 (x-1)dx = [x^2/2 - x]_1^2 = 1/2
∫_2^3 xp(x)dx = ∫_2^3 (x-1)dx = [x^2/2 - x]_2^3 = 3/2
∫_3^6 xp(x)dx = ∫_3^6 (1/3)dx = 1
∫_6^10 xp(x)dx = ∫_6^10 (x/12)dx = [x^2/24]_6^10 = 5/2
Therefore, we have
E(X) = 0 + 1/2 + 3/2 + 1 + 5/2 + 0 = 8
So the expected value of X is 8.
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