A particular fruit's weights are normally distributed, with a mean of 426 grams and a standard deviation of 37 grams. If you pick 9 fruit at random, what is the probability that their mean weight will be between 413 grams and 464 grams. Round to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 2 decimal places are accepted.

Final answer:

The function in question, f(x) = 3x^2 + 7x + 2, is a quadratic function, and its properties such as graph shape, intercepts, and vertex can be studied. Additionally, the derivative of this function, obtained through power rule differentiation, is f'(x) = 6x + 7.

Explanation:

The question asks about the function f(x) = 3x2 + 7x + 2. This appears to be a quadratic function, which is a fundamental concept in algebra and pre-calculus. Detailing the characteristics of a quadratic function involves finding its graph, which is a parabola, its vertex, axis of symmetry, intercepts, and possibly its extrema (maximum or minimum values).

In mathematics, finding the derivative of a function is a common operation in calculus. Given the information on different functions and their derivatives from the provided reference text, we can deduce that the derivative of f(x) would be found through power rule differentiation: for f(x) = axn, the derivative f'(x) = naxn-1. Applying this to the given function, we find the derivative f'(x) = 6x + 7.

The equivalent expression to 2 (three-fourths x + 7) minus 3 (one-half x minus 5) is 29.

To find equivalent expressions to 2 (three-fourths x + 7) minus 3 (one-half x minus 5), we first need to distribute the numbers outside the parentheses inside the parentheses. This simplifies the expression as follows:

Subtracting the second expression from the first, we obtain: (1.5x + 14) - (1.5x - 15), which simplifies to 0x + 29, or simply 29.

Therefore, the only expression equivalent to the given expression is 29.

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Answer:

The answer is B, C, and E

Step-by-step explanation:

Answer:

a

Step-by-step explanation:

The order of four steps are Assessing needs and wants, Considering opportunity costs, Assessing risks and returns and Setting short- and long-term goals, Option C is correct.

Finance is the study and discipline of money, currency and capital assets.

The correct order of the four steps in solving one's personal financial challenges is indeed:

Assessing needs and wants

Considering opportunity costs

Assessing risks and returns

Setting short- and long-term goals

Hence, the order of four steps are Assessing needs and wants, Considering opportunity costs, Assessing risks and returns and Setting short- and long-term goals, Option C is correct.

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Answer: y = 0.25*f(x - A) + 3

Step-by-step explanation:

Initially we have the graph of y = f(x)

If we do a vertical compression, this means that we multiply the function by the scale factor, in this case the scale factor is 0.25

So now our graph is y = 0.25*f(x)

A translation to the right by A units means that now we valuate the function in x - A, in this case A = 4, so our graph now is:

y = 0.25*f(x - 4)

A vertical translation means that we add a constant to the function, if the constant is positive the tranlsation is upwards, if the constant is negative the translation is downwards.

Here the translation is of 3 units upwards, so our new graph is:

y = 0.25*f(x - A) + 3

This problem involves conducting a hypothesis test, stating the null hypothesis as 78% or fewer readers own a personal computer, and the alternative hypothesis as more than 78% own one. A statistical test (like a z-test) is needed to compare the p-value to the 0.01 significance level. The publisher's claim is supported if the p-value is less than 0.01.

In this context, we are conducting a hypothesis test to examine a newsletter publisher's claim about their audience's computer ownership. The null and alternative hypotheses would therefore be set up as follows:

Null Hypothesis (H0): p <= 0.78, meaning 78% or fewer readers own a personal computer.

Alternative Hypothesis (Ha): p > 0.78, meaning more than 78% of readers own a personal computer.

We can conduct a hypothesis test to prove or disprove this using statistical methods such as a z-test and by comparing the p-value to the significance level (0.01 in this case). If the p-value obtained from the test is less than 0.01, we can reject the null hypothesis providing sufficient evidence to support the publisher's claim. If not, we fail to reject the null hypothesis which means the evidence is insufficient.

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Answer:

The solution is shown in the picture attached

Step-by-step explanation:

To verify X as a solution to the system, substitute X into the system and check for equality. X is a vector and X' a matrix. Careful calculations with matrices and trigonometric identities are necessary for the verification.

To verify that the vector X is indeed a solution to the given system, we can substitute X into the system and check if both sides are equal. If they are, then X is a solution to the system.

X in this case is a vector whose elements are trigonometric functions of time t. Likewise, X' represents a matrix multiplying the vector X. After applying the multiplication, we can compare the resulting vector with the original vector.

As this involves calculation with matrices and trigonometric identities, careful execution of these steps is necessary to ensure the accuracy of the result.

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Answer:

Step-by-step explanation

First five multiples of 2030 are: 2, 5, 7, 10 an 14

Answer:

  √37

Step-by-step explanation:

It is helpful to know the squares of small integers. Then you become aware of the approximate magnitudes of square roots.

Point C is between 6 and 7, so the value of C² will be between 6² = 36 and 7² = 49. Since C is closer to 6 than to 7, it represents the root of a number closer to 36 than to 49.

Only one answer choice is in this range: √37, Option 1.

Answer: A. have the same slope

Step-by-step explanation:

b. bisect means right in half but not fully intersecting kinda looks like this _l_ *oh and it asked you to prove that they bisect so clicking on it don't really make sense*

c. perpendicular means there is four equal angels and that is 90 degree angels

d. parallel... well it's obviously not parallel because parallel are two lines that are exactly the same but never intersect

*intersect means touch*

Answer:

9

Step-by-step explanation:

In order to figure this out, I made the proportion:

[tex]\frac{300}{x} =\frac{100}{3}[/tex]

Multiply 3 by 300 and x by 100

300*3=?

x*100=?

900=100x

Divide 900 by 100 to find x

900÷100=9

x=9

Answer:

Step-by-step explanation:

WE are given that [tex]a_n = \frac{(-1)^{n+1}}{5n-4}[/tex]. Then, to now the first for terms, we must replace n by 1,2,3,4 respectively. Then

[tex]a_1 = \frac{(-1)^2}{5(1)-4} = \frac{1}{1}= 1 [/tex]

[tex]a_2 = \frac{(-1)^3}{5(2)-4} = \frac{-1}{6} [/tex]

[tex]a_3 = \frac{(-1)^4}{5(3)-4} = \frac{1}{11}= 1 [/tex]

[tex]a_4 = \frac{(-1)^5}{5(4)-4} = \frac{-1}{16}= 1 [/tex]

Note that as n increase, [tex]a_n[/tex] gets closer to 0. So, the limit of this sequence is 0.

Answer: The answer is p = 343

Given that 'p' is inversely proportional to the square of 'q', we first found the constant of proportionality (k) by substituting the given 'p' and 'q' values. With 'k' known, we substituted the new value of 'q' to find the corresponding value of 'p', which turned out to be 343 when q=2.

The given question describes an inverse proportionality. Specifically, it states that p is inversely proportional to the square of q. To express this mathematically, we write it as p = k/(q^2), where k is the constant of proportionality. For finding this constant, we use the given values of p and q, so 28 = k/(7^2), which means k = 28*49 = 1372.

Now, we substitute the value of k and the new value of q into the equation to find the corresponding value of p. Hence, when q = 2, p = 1372/(2^2) = 1372/4 = 343. Therefore, when q = 2, p equals 343.

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Answer:

x+9

Step-by-step explanation:

1/3x[8-5]+9=1/3x(3)+9

1/3x(3)+9=x+9

x+9

The final result of the expression 1/3x[8-5]+9 is 10.

To evaluate the expression 1/3x[8-5]+9, we need to perform the operations in the correct order, following the PEMDAS/BODMAS rule. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS is similar but is used in some regions with the 'O' representing "Orders" or "Indices" instead of Exponents.

Step 1: Inside the brackets, we solve the expression 8 - 5, which gives us 3.

Step 2: Now, we have 1/3x3 + 9. According to PEMDAS/BODMAS, we must perform multiplication before addition. So, we proceed with 1/3 times 3, which equals 1.

Step 3: The expression now simplifies to 1 + 9.

Step 4: Finally, we perform the addition, which yields 10.

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Answer:

17

Step-by-step explanation:

Let x, x +1, x + 2, x + 3 be four consecutive integers.

[tex] \therefore \: x + x +1 + x + 2 + x + 3 = 74 \\ \therefore \: 4x + 6 = 74 \\ \therefore \: 4x = 74 - 6 \\ \therefore \: 4x = 68 \\ \therefore \: x = \frac{68}{4} \\ \huge \red{ \boxed{x = 17}}[/tex]

Hence, first integer is 17.

Answer:

17 is your answer

hope this helps :)

Step-by-step explanation:

Answer:

The answer is d (13.26)

Step-by-step explanation:

set it up like this: (95/360) times (2 times pi times 8)

after plugging this equation into a calculator you get 13.26450232 and round to 13.26

Final answer:

The arc length of a circle with an 8-inch radius and a central angle of 95 degrees is 13.26 inches when we use 3.14 for π and round to the nearest hundredth.

Explanation:

To find the arc length of a circle, we use the formula arc length (Δs) = rΘ, where 'r' is the radius and Θ is the central angle in radians. Since there are 2π radians in a full 360-degree rotation, we can find the radian measure of 95 degrees by using the conversion ratio π radians/180 degrees. The radian measure is (95/180)π.

Using 3.14 for π and the given radius of 8 inches, the calculation becomes: Δs = 8 * (95/180) * 3.14. Simplifying this equation gives the arc length as 13.26 inches when rounded to the nearest hundredth

Answer:

55 cm2

Step-by-step explanation:

The area of the new parallelogram is 55 sq.cm, the correct option is C.

A polygon with four sides such that the opposite sides are parallel and equal is called a Parallelogram.

The height of the parallelogram is 4 cm

The base of the parallelogram is 9cm

The height of the parallelogram is increased by 1 cm

New height = 5cm

The base of the parallelogram is increased by 2 cm

New base = 11 cm

Area of a parallelogram is =  Base * Height

Area of parallelogram is  = 5 * 11 = 55 sq.cm

Therefore, the area of the new parallelogram is 55 sq.cm.

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Answer: The answer is y = 18(1.15)^x

Step-by-step explanation:

Answer:

y = (18) * (1.15)^x

Step-by-step explanation:

A limited edition poster increases in value each year with an initial value of $18. After 1 year and an increase of 15% per year, the poster is worth $20. 70. Which equation can be used to find the y value after x years.

To find this equation, this is an exponential equation, meaning that the number increases at a rapid rate. In this case the post increases each year by 15% and started at $18. The equation you would use to find this is: y = a * b^x. We can fill in the a and b values based off the given information. Since the value is increasing by 15% we will add 1 to 15% to get 1.15. This will be the b value. The a value is our initial value which in this case is $18. Now we can plug everything in to get: y = (18) * (1.15)^x.

Answer:

a) The volume of S is 34.13

b) The volume of S is 14.8

c) The volume of S is 5.17

d) The volume of S is 11.33

Step-by-step explanation:

a) The cross section area is equal to:

[tex]A=a^{2} =((5x-x^{2})-x)^{2} =(4x-x^{2} )^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^4_0 {A(x)} \, dx =\int\limits^4_0 {(4x-x^{2})^{2} } \, dx =34.13[/tex]

b) The cross section area is equal to:

[tex]A=\frac{a^{2}\sqrt{3} }{4} =\frac{\sqrt{3} }{4} ((5x-x^{2} )-x)^{2} =\frac{\sqrt{3} }{4} (4x-x^{2} )^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^4_0 {A(x)} \, dx =\frac{\sqrt{3} }{4} \int\limits^4_0 {(4x-x^{2})^{2} } \, dx =14.8[/tex]

c)

[tex]y=5x-x^{2} \\\frac{dy}{dx} =0\\5x-x^{2} =0\\x=5/2\\y(5/2)=25/4\\y=5x-x^{2} \\x^{2} -5x+y=0\\x=\frac{5+-\sqrt{25-4y} }{2}[/tex]

The cross section area is equal to:

[tex]A_{1} =\frac{1}{2} \pi r_{1}^{2} =\frac{1}{2} \pi (\frac{1}{2} (\frac{5+\sqrt{25-4y} }{2} -\frac{5-\sqrt{25-4y} }{2} ))^{2} =\frac{1}{8} \pi (25-4y)\\A_{2} =\frac{1}{2} \pi r_{2}^{2}=\frac{1}{2}\pi (\frac{1}{2} (y-\frac{5-\sqrt{25-4y} }{2} ))^{2} =\frac{1}{32} \pi (2y-5+\sqrt{25-4y} )^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^a_b {A_{1}(y) } \, dy+\int\limits^4_0 {A_{2}(y) } \, dy ,where-a=25/4,b=4\\Vol_{S} =\int\limits^a_b {\frac{1}{8}\pi (25-4y)} \, dy +\int\limits^a_b {\frac{1}{32}\pi (2y-5+\sqrt{25-4y} )^{2} } \, dy =5.17[/tex]

d) The cross section area is:

[tex]A_{1} =\frac{1}{2}ab=\frac{1}{2} a^{2} =\frac{1}{2} (\frac{5+\sqrt{25-4y} }{2}-\frac{5-\sqrt{25-4y}}{2} )^{2} =\frac{1}{2} (25-4y)\\A_{1}=\frac{1}{2}ab=\frac{1}{2} a^{2} =\frac{1}{2}(y-\frac{5-\sqrt{25-4y}}{2}} )^{2} =\frac{1}{8} (2y-5+\sqrt{25-4y}})^{2}[/tex]

The volume of S is equal to:

[tex]Vol_{S} =\int\limits^a_b {A_{1}(y) } \, dy +\int\limits^4_0 {A_{2}(y) } \, dy ,where-a=25/4,b=4\\Vol_{S}=\int\limits^a_b {\frac{1}{2}(25-4y) } \, dy +\int\limits^4_0 {\frac{1}{8}(2y-5+\sqrt{25-4y})^{2} } \, dy =11.33[/tex]

A circle with radius r has a circumference of 2πr.

2πr = 18.84  ==>  r = 18.84/(2π) ≈ 2.998 units

The same circle has area πr ² sq. units. So the area is

π (2.998)² ≈ 28.246 sq. units

To find the area of a circle with a given circumference, divide the circumference by 2π to find the radius. Then, plug the radius into the formula A = πr^2 to calculate the area.

To find the area of a circle when given the circumference, you can use the formula A = πr^2.

First, determine the radius of the circle by dividing the circumference by 2π.

In this case, the circumference is 18.84, so the radius would be 18.84 / (2π) = 3.0.

Then, plug the radius into the formula A = πr^2 to find the area:

A = π(3.0)^2 = 9.0π = 28.27 (rounded to two decimal places)

Therefore, the area of the circle with a circumference of 18.84 units is approximately 28.27 square units.

Answer:

a) If Y is distributed N(1, 4), Pr (Y ≤ 3) = 0.84134

b) If Y is distributed N(3, 9), Pr (Y > 0) = 0.84134

c) If Y is distributed N(50, 25), Pr (40 ≤ Y ≤ 52) = 0.63267

d) If Y is distributed N(5, 2), find Pr (6 ≤ Y ≤ 8) = 0.22185

Step-by-step explanation:

With the logical assumption that all of these distributions are normal distribution,

a) Y is distributed N(1, 4), find Pr ( Y ≤ 3 )

Mean = μ = 1

Standard deviation = √(variance) = √4 = 2

To find the required probability, we first standardize 3

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (y - μ)/σ = (3 - 1)/2 = 1

We'll use data from the normal probability table for these probabilities

The required probability

Pr ( Y ≤ 3 ) = P(z ≤ 1) = 0.84134

b) If Y is distributed N(3, 9), find Pr ( Y > 0 )

Mean = μ = 3

Standard deviation = √(variance) = √9 = 3

To find the required probability, we first standardize 0

The standardized score for any value is the value minus the mean then divided by the standard deviation.

z = (y - μ)/σ = (0 - 3)/3 = -1

We'll use data from the normal probability table for these probabilities

The required probability

Pr ( Y > 0) = P(z > -1) = 1 - P(z ≤ -1) = 1 - 0.15866 = 0.84134

c) If Y is distributed N(50, 25), find Pr (40 ≤ Y ≤ 52).

Mean = μ = 50

Standard deviation = √(variance) = √25 = 5

To find the required probability, we first standardize 40 and 52.

For 40,

z = (y - μ)/σ = (40 - 50)/5 = -2

For 52,

z = (y - μ)/σ = (52 - 50)/5 = 0.4

We'll use data from the normal probability table for these probabilities

The required probability

Pr (40 ≤ Y ≤ 52) = P(-2.00 ≤ z ≤ 0.40)

= P(z ≤ 0.40) - P(z ≤ -2.00)

= 0.65542 - 0.02275

= 0.63267

d) If Y is distributed N(5, 2), find Pr ( 6 ≤ Y ≤ 8 )

Mean = μ = 5

Standard deviation = √(variance) = √2 = 1.414

To find the required probability, we first standardize 6 and 8.

For 6,

z = (y - μ)/σ = (6 - 5)/1.414 = 0.71

For 8,

z = (y - μ)/σ = (8 - 5)/1.414 = 2.12

We'll use data from the normal probability table for these probabilities

The required probability

Pr (6 ≤ Y ≤ 8) = P(0.71 ≤ z ≤ 2.12)

= P(z ≤ 2.12) - P(z ≤ 0.71)

= 0.983 - 0.76115

= 0.22185

Hope this Helps!!

Answer:

a) The value of N(1, 4) = 0.8413

b) The probability of N(3, 9) = 0.8413

ci) The probability (40≤ Y≤ 52) = 0.4

cii) The probability of N (3, 9) = 0.6236

d) The probability of (6≤Y≤8) = 0.2216

Step-by-step explanation:

Detailed step by step explanation is given in the attached document.

A normal distribution is a bell shaped symmetric distribution. This kind of distribution has a normal probability density function. A standard normal distribution is the one that has a mean 0 and variance of 1. It is often denoted as N (0, 1). If a general variance and mean are given and one has to look up probabilities in a normal probability distribution. The variable is standardized first. Standardizing a variable involves subtracting the general mean from the standard and then dividing the result by 1. In order to find the probabilities, the value of z is located in a normal distribution table.

Answer:

The solution to the differential equation y'(y + 7)y'' = (y')²

y = Ae^(Kx) - 7

Step-by-step explanation:

Given the differential equation

y'(y + 7)y'' = (y')² ..................(1)

We want to solve using the substitution u = y'.

Let u = y'

The u' = y''

Using these, (1) becomes

u(y + 7)u' = u²

u' = u²/u(y + 7)

u' = u/(y + 7)

But u' = du/dy

So

du/dy = u/(y + 7)

Separating the variables, we have

du/u = dy/(y + 7)

Integrating both sides, we have

ln|u| = ln|y + 7| + ln|C|

u = e^(ln|y + 7| + ln|C|)

= K(y + 7)

But u = y' = dy/dx

dy/dx = K(y + 7)

Separating the variables, we have

dy/(y + 7) = Kdx

Integrating both sides

ln|y + 7| = Kx + C1

y + 7 = e^(Kx + C1) = Ae^(Kx)

y = Ae^(Kx) - 7

Final answer:

To solve the given differential equations by using the substitution u = y', substitute u for y' and find the values of u. Then, solve the resulting first order ordinary differential equation by separating variables and integrating to determine the solution.

Explanation:

To solve the given differential equations by using the substitution u = y', we need to substitute u for y' and find the values of u. Let's take the first equation as an example:

Start by substituting u for y' in the equation: (y + 7)y'' = (y')^2

Replace y' with u in the equation: (y + 7)u' = u^2

Then, we can solve this first order ordinary differential equation by separating variables and integrating:

Divide both sides by (y + 7): u' = (u^2) / (y + 7)

Separate the variables: (y + 7)dy = (u^2)du

Integrate both sides: (1/2)(y^2 + 14y) = (1/3)u^3 + C (where C is the constant of integration)

Solve for y by rearranging the equation: y^2 + 14y = (2/3)u^3 + 2C

This is the solution to the given differential equation.

Answer:

a) [tex]t=\frac{(56.2-64.2)-0}{\sqrt{\frac{18.2^2}{22}+\frac{13.9^2}{11}}}}=-1.40[/tex]  

b) [tex]p_v =2*P(t_{31}<-1.4)=0.171[/tex]  

Step-by-step explanation:

Information given

[tex]\bar X_{1}=56.2[/tex] represent the mean for sample 1  

[tex]\bar X_{2}=64.2[/tex] represent the mean for sample 2  

[tex]s_{1}=18.2[/tex] represent the sample standard deviation for 1  

[tex]s_{2}=13.9[/tex] represent the sample standard deviation for 2  

[tex]n_{1}=22[/tex] sample size for the group 2  

[tex]n_{2}=11[/tex] sample size for the group 2  

t would represent the statistic (variable of interest)  

System of hypothesis

We need to conduct a hypothesis in order to check if the true means are different, the system of hypothesis would be:  

Null hypothesis:[tex]\mu_{1}-\mu_{2}=0[/tex]  

Alternative hypothesis:[tex]\mu_{1} - \mu_{2}\neq 0[/tex]  

The statistic is given by:

[tex]t=\frac{(\bar X_{1}-\bar X_{2})-\Delta}{\sqrt{\frac{s^2_{1}}{n_{1}}+\frac{s^2_{2}}{n_{2}}}}[/tex] (1)  

The degrees of freedom are given by:

[tex]df=n_1 +n_2 -2=22+11-2=31[/tex]  

Part a: Statisitc

Replacing into the formula we got:

[tex]t=\frac{(56.2-64.2)-0}{\sqrt{\frac{18.2^2}{22}+\frac{13.9^2}{11}}}}=-1.40[/tex]  

Part b: P value  

The p value on this case would be:

[tex]p_v =2*P(t_{31}<-1.4)=0.171[/tex]  

Answer:

Step-by-step explanation:

2: 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42

3: 3 6 9 12 15 18 21 24 27 30 33 36 39 42

7: 7 14 21 28 35 42

Answer:

120

Step-by-step explanation:

add 4 and 2 then multipy by 2 and add a zero

Answer:

y = 1/2x -1

Step-by-step explanation:

to find the y-intercept you find where the line crosses the y axis

for the slope you need to use the equation

change of y2 - change of y1

change of x2 - change of x1

y=ax+b

(-2; -2); (0;-1); (2; 0)

-1=a*0+b => b=-1

0=a*2+b

0=a*2-1

a*2=1 => a=1/2

y=x/2 -1

Answer:

0.069 = 6.9% probability that a customer has to wait more than 4 minutes.

Step-by-step explanation:

Exponential distribution:

The exponential probability distribution, with mean m, is described by the following equation:

[tex]f(x) = \mu e^{-\mu x}[/tex]

In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.

The probability that x is lower or equal to a is given by:

[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]

Which has the following solution:

[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]

In this problem, we have that:

[tex]m = 1.5[/tex]

So

[tex]\mu = \frac{1}{1.5} = 0.6667[/tex]

[tex]P(X \leq x) = 1 - e^{-0.667x}[/tex]

Find the probability that a customer has to wait more than 4 minutes.

Either the customer has to wait 4 minutes or less, or he has to wait more than 4 minutes. The sum of the probabilities of these events is decimal 1. So

[tex]P(X \leq 4) + P(X > 4) = 1[/tex]

We want P(X > 4). So

[tex]P(X > 4) = 1 - P(X \leq 4) = 1 - (1 - e^{-0.667*4}) = 0.069[/tex]

0.069 = 6.9% probability that a customer has to wait more than 4 minutes.

The microphone, which is at the focal point of a parabolic microphone, should be placed 5 inches from the vertex of the parabola since the parabola is 40 inches wide and 20 inches deep.

The question deals with a parabolic shape and requires using the properties of a parabola to determine the position of the focal point. In the context of a parabolic microphone, the focal point is where the microphone should be placed to best capture sounds. For a parabola given in the form y = 4px, where p is the distance from the vertex to the focus, and the parabola is 40 inches wide (which is the distance from one end of the parabola to the other at the given depth) and 20 inches deep (which is the distance from the vertex to the directrix), we can find p, using the relationship depth = p. Since the depth is 20 inches, the focus (and thus where the microphone should be placed) is 20/4 = 5 inches from the vertex of the parabola.

Answer:

a) The distance from the television camera to the rocket is changing at that moment at a speed of

600 ft/s

b) the camera's angle of elevation is changing at that same moment at a rate of

0.16 rad/s = 9.16°/s

Step-by-step explanation:

This is a trigonometry relation type of problem.

An image of when the rocket is 3000 ft from the ground is presented in the attached image.

Let the angle of elevation be θ

The height of the rocket at any time = h

The distance from the camera to the rocket = d

a) At any time, d, h and the initial distance from the camera to the rocket can be related using the Pythagoras theorem.

d² = h² + 4000²

Take the time derivative of both sides

(d/dt) (d²) = (d/dt) [h² + 4000²]

2d (dd/dt) = 2h (dh/dt) + 0

At a particular instant,

h = 3000 ft,

(dh/dt) = 1000 ft/s

d can be obtained using the same Pythagoras theorem

d² = h² + 4000² (but h = 3000 ft)

d² = 3000² + 4000²

d = 5000 ft

2d (dd/dt) = 2h (dh/dt) + 0

(dd/dt) = (h/d) × (dh/dt)

(dd/dt) = (3000/5000) × (1000)

(dd/dt) = 600 ft/s

b) If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at that same moment?

At any moment in time, θ, h and the initial distance of the camera from the base of the rocket are related through the trigonometric relation

Tan θ = (h/4000) = 0.00025h

Taking the time derivative of both sides

(d/dt) (Tan θ) = (d/dt) (0.00025h)

(Sec² θ) (dθ/dt) = 0.00025 (dh/dt)

At the point where h = 3000 ft, we can calculate the corresponding θ at that point

Tan θ = (3000/4000)

θ = tan⁻¹ (0.75) = 0.6435 rad

(Sec² θ) (dθ/dt) = 0.00025 (dh/dt)

(Sec² 0.6435) (dθ/dt) = 0.00025 (1000)

1.5625 (dθ/dt) = 0.25

(dθ/dt) = (0.25/1.5625) = 0.16 rad/s

Hope this Helps!!!

The rate of change of the hypotenuse distance and the camera's angle of elevation can be calculated using the principles of trigonometry and differentials. The rates derive from the Pythagorean theorem and the derivatives of trigonometric functions, respectively.

This question relates to the concepts of trigonometry and differential calculus. We can see the camera, the rocket and the launch pad as forming a right triangle: the distance from the camera to the rocket is the hypotenuse, the distance from the camera to the launch pad is one leg (adjacent to the angle of elevation) and the distance that the rocket has risen is the other leg (opposite to the angle of elevation).

(a) To find how fast the distance from the camera to the rocket is changing, we can use the Pythagorean theorem (a² + b² = c²). Here, a = 4000 ft, b = 3000 ft, so, c = sqrt((4000)² + (3000)²). The derivative dc/dt (rate of change of c) when b = 3000 ft and db/dt = 1000 ft/s will provide the answer.

(b) To find how fast the camera's angle of elevation (let's symbolize it by θ) is changing, we use the concept of derivatives of trigonometric functions, specifically the tangent, which is defined as opposite (b) over adjacent (a), or tan(θ) = b/a. Then, we can compute dθ/dt using implicit differentiation when b = 3000 ft and db/dt = 1000 ft/s.

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Answer:

[tex]\$456[/tex]

Step-by-step explanation:

The correct question is

Mr. Hopkins is building a sandbox for his children. It would cost $228 for the sand if he builds a sandbox with dimensions 9 ft by 6 ft. If Mr. Hopkins decides to increase the size to 13 1/2 ft by 8 ft but keep the same depth of sand, how much would the sand cost

step 1

Find the unit rate of the cost per square foot

Divide the total cost by the area

[tex]\frac{228}{9*6}=\$\frac{38}{9}\ per\ square\ foot[/tex]

step 2

Find the area of the increased sandbox

we have

[tex]L=13\frac{1}{2}=13.5\ ft[/tex]

[tex]W=8\ ft[/tex]

The area is equal to

[tex]A=(13.5)(8)=108\ ft^2[/tex]

step 3

Find the cost of the new sandbox

Multiply the area by the cost per square foot

[tex]108(\frac{38}{9})=\$456[/tex]

Answer:

12

Step-by-step explanation: