what is the approximate volume of the cylinder? use 3.14 for TT 7 cm radius 26 cm height

Answer:

a) The Venn diagram is presented in the attached image to this answer.

b) 0.82

c) 0.16

Step-by-step explanation:

a) The Venn diagram is presented in the attached image to this answer.

n(U) = 100%

n(S) = 48%

n(B) = 66%

n(H) = 38%

n(S n B) = 30%

n(B n H) = 22%

n(S n H) = 28%

n(S n B n H) = 12%

The specific breakdowns for each subgroup is calculated on the Venn diagram attached.

b) The probability that a randomly selected student likes basketball or hockey.

P(B U H)

From the Venn diagram,

n(B U H) = n(S' n B n H') + n(S' n B n H) + n(S n B n H') + n(S n B n H) + n(S n B' n H) + n(S' n B' n H) = 26 + 10 + 18 + 12 + 16 + 0 = 82%

P(B U H) = 82/100 = 0.82

c) The probability that a randomly selected student does not like any of these sports.

P(S' n B' n H')

n(S' n B' n H') = n(U) - [n(S' n B n H') + n(S' n B n H) + n(S n B n H') + n(S n B n H) + n(S n B' n H) + n(S' n B' n H) + n(S n B' n H')]

n(S' n B' n H') = 100 - (26 + 10 + 18 + 12 + 16 + 0 + 2) = 100 - 84 = 16%

P(S' n B' n H') = 16/100 = 0.16

Answer:

about 3 1/2 hours long

Step-by-step explanation:

70 goes into 240 evenly 3 times, and 70×3 equals 210. that leaves 30 miles remaining. since he is moving at an average rate of 70mph, 3/7 would be the remaining time. to decimal that equals a total of about 3 hours and 25 minutes, which, rounded up, equals 3 and 1 half hours.

Answer:

This system have 9000 different numbers.

Step-by-step explanation:

We know that the internal telephone numbers in the phone system on a campus consists of 4 digits, with the 1st not equal to 0.

We have total 10 digits.

In the first place we have 9 possibilities, because from the conditions of the task in the first place there cannot be 0.

In second, third and fourth place we have 10 possibilities.

Therefore, we get

[tex]N=9\cdot 10\cdot10 \cdot 10=9000[/tex]

This system have 9000 different numbers.

In this phone system, there are 9,000 different internal telephone numbers that can be assigned.

The internal telephone numbers in the phone system on a campus consists of 4 digits, with the 1st not equal to 0. To find the number of different numbers that can be assigned in this system, we need to consider the possibilities for each digit position. Since the first digit cannot be 0, there are 9 choices (1 through 9) for the first digit. For the remaining three digits, each digit can be any number from 0 to 9, resulting in 10 choices for each of the three remaining digits. Therefore, the total number of different numbers that can be assigned in this system is

9 * 10 * 10 * 10 = 9,000.

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Answer:

y = 2x^2 - 12x + 27

Step-by-step explanation:

Step 1:  Distribute the power

y = 2(x - 3)² + 9

y = 2(x^2 - 6x + 9) + 9

y = 2x^2 - 12x + 18 + 9

y = 2x^2 - 12x + 27

Answer: y = 2x^2 - 12x + 27

Answer:

  -540

Step-by-step explanation:

The coefficient of the k-th term of (a+b)^n is nCk. (k = 0, 1, 2, ..., n)

Here, we want the coefficient of a^3b^3 for n=6, so k=3 and the coefficient is ...

  6C3 = 6!/(3!(6-3)!) = 5·4 = 20

So for a=3x and b=-y, we want ...

  20a^3b^3 = 20(3x)^3(-y)^3 = -540x^3y^2

The coefficient you're looking for is -540.

Answer: the amount of interest earned is $44.7

Step-by-step explanation:

The formula for determining simple interest is expressed as

I = PRT/100

Where

I represents interest paid on the amount deposited.

P represents the principal or amount deposited.

R represents interest rate

T represents the duration in years.

From the information given,

P = $1295

R = 7%

T = 180 day. Assuming there are 365 days in a year. Converting 180 days to years, it becomes

180/365 = 0.49315 year

Therefore,

I = (1295 × 7 × 0.49315)/100 = $840,

I = $44.7

Answer:

Step-by-step explanation:

In a parallelogram, the opposite sides are equal and parallel.

Also, consecutive angles in a parallelogram are supplementary. This means that the sum of the consecutive angles is 180 degrees.

Angle 2m and angle n are supplementary angles. Therefore,

2m + n = 180 - - - - - - - - - - - -1

Also, the opposite angle in a parallelogram are equal. This means that

2m = 70

m = 70/2

m = 35 degrees

Substituting m = 35 into equation 1, it becomes

2 × 35 + n = 180

70 + n = 180

n = 180 - 70

n = 110 degrees

The value of m and n are 35 and 110 respectively.

We are given that one of the angles of the parallelogram is [tex]70^\circ[/tex].

Since opposite angles in a parallelogram are supplementary, the angle directly opposite the 70° angle must be [tex]180^\circ -70^\circ = 110^\circ[/tex].

We are also given that one of the non-consecutive interior angles has a measure of 2m°.

Since consecutive interior angles in a parallelogram add up to 180°, the angle next to the 2m° angle must have a measure of 180°- 2m°.

The opposite sides of a parallelogram are congruent.

This means that n° must be congruent to 110°. So, n = 110.

We can now solve for 'm'.

Since the angle next to the 2m° angle must have a measure of 180°−2m° and this angle is also congruent to 110°.

We have the equation: [tex]110^\circ = 180^\circ - 2m^\circ[/tex]

[tex]2m^\circ = 70^\circ[/tex]

[tex]m=35[/tex]

Answer:

(a)His monthly Interest Rate=0.8%

(b)Annual Interest Rate = 9.6%

(c)[tex]500(1.008)^m[/tex]

Step-by-step explanation:

For a Principal P invested at a yearly rate r, compounded m times in t years

Amount at Compound Interest= [tex]P(1+\frac{r}{m})^{mt}[/tex]

Comparing with Jerry's equation y=388 (1.008)

(a)His monthly Interest Rate= 0.008=0.8%

(b)Annual Interest Rate= Monthly Interest Rate X 12 =0.8 X 12 = 9.6%

(c)If I invest $500 at the same rate of return,

Total Money after m months

= [tex]P(1+\frac{r}{m})^{mt}[/tex][tex]=500(1+0.008)^{m}[/tex][tex]=500(1.008)^m[/tex]

Final answer:

Jerry's monthly interest rate is 0.8%, and his annual interest rate is approximately 9.96%. An investment of $500 in the same account would grow according to the equation [tex]y = 500(1.008)^{m}[/tex], where m represents the months invested.

Explanation:

Jerry says he's got his money in a great account that compounds interest monthly. The equation [tex]y=388(1.008)^{m}[/tex] represents how much money he has at the end of the month. The monthly interest rate is found in the equation inside the parentheses, 1.008, which means the monthly interest rate is 0.8% (since 1.008 is equal to 1 plus 0.008 or 1 + 0.8/100). To find the annual interest rate, we need to use the compounding formula, which for monthly compounding can be simplified to [tex](1 + monthly interest rate)^{12} - 1[/tex]. so, [tex](1.008)^{12}- 1[/tex] gives an annual rate of approximately 9.96%.

To write an equation representing your total money if you invest $500 in an account with the same rate of return, we use the formula [tex]y = P(1 + r)^{m}[/tex], where P is the principal amount ($500), r is the monthly interest rate (0.008), and m represents the number of months the money has been invested. Therefore, the equation is [tex]y = 500(1.008)^{m}[/tex].

The mean and standard deviation for the Standard normal distribution is 0 and 1 respectively. Option b is correct.

The standard normal distribution is a type of normal distribution that has a mean of 0 and a standard deviation of 1. The standard deviation shows how much a particular measurement deviates from the mean, and the standard normal distribution is centered at zero.

Since the standard normal distribution is a normal distribution curve in which the values of the mean is 0 and the value of The standard normal distribution is a normal distribution curve where the values of the mean and standard deviation is 1.

Thus, the correct option for the given question is Option B.

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In a standard normal distribution, the mean is 0 and the standard deviation is 1. This defines a distribution where data is symmetrically spread around the mean, and a z-score can be used to determine how many standard deviations a value is from the mean.

In a standard normal distribution, the correct answer to the student's question is that the mean is 0 and the standard deviation is 1. This is represented by option b. In a standard normal distribution, often denoted as Z ~ N(0, 1), the mean (μ) equals 0, which signifies that the distribution is centered on the zero point on a number line.

The standard deviation (σ) equals 1, indicating that the values within the distribution are spread out in such a way that one standard deviation away from the mean encompasses approximately 68% of the data in a symmetrical fashion on both sides of the mean.

The concept of z-scores in the context of the standard normal distribution allows for the comparison of different values within different populations. A z-score represents how many standard deviations away a value is from the population mean.

It is crucial to note that the standard deviation cannot be negative; it represents the dispersion of the dataset around the mean, and therefore can only be positive or zero.

Answer:

30

Step-by-step explanation:

Look at the bottom line of the chart.

He sold 41 sides of French fries and 40 sides of onion rings.

41 + 40 = 81

Since he sold a total of 81 sides with bacon cheeseburgers, that means he sold a total of 81 bacon cheeseburgers.

We have a ratio:

81 bacon cheeseburgers to 40 sides of onion rings

Now he sold 120 bacon cheeseburgers, so we set up a proportion to find the number of sides of onion rings. Let the unknown number be x.

81 burgers is to 40 sides as 120 burgers is to x sides

81/40 = 120/x

We solve for x by cross multiplying.

81x = 40 * 120

81x = 4800

x = 4800/81 = 59.3

That means he serves 59 sides of onion rings.

Each serving of onion rings is half pound.

59 * 0.5 = 29.5

For 120 bacon cheeseburgers, he serves approximately 29.5 lb of onion rings.

Answer: 30

Answer:

30 pounds

Bacon cheeseburger with onion rings:

40

(41 + 40)

=

40

81

= 0.4938

then,

120 x 0.4938 = 59.256

then,

59.256 x .50 = 29.628

Step-by-step explanation:

Below is an attachment containing the solution.

Final answer:

The dimensions are 24cm x 15cm x 20cm.

Explanation:

To find two possible sets of lengths and widths for the rectangular prism with a height of 20cm and a surface area of 2400cm2, we need to understand the formula for the surface area of a rectangular prism: SA = 2(lw + lh + wh). Here, l = length, w = width, and h = height.

Since we know the height (h = 20cm) and the surface area (SA = 2400cm2), we can set up the equation:

2(lw + 20l + 20w) = 2400

We will consider two cases: one where the length and width are equal (since that is a specific request), and another where they are not.

Case 1: Length and width are equal (l = w). We simplify the equation to:

2(l2 + 40l) = 2400

Solving for l gives us l = w = 20cm. Therefore, the dimensions are 20cm x 20cm x 20cm.

Case 2: Length and width are not equal. To find a possible set, we can assume a width and solve for the length:

Let's assume w = 15cm. Plugging this into the equation gives us:

2(l15 + 20l + 20 x 15) = 2400

Solving for l gives us l = 24cm.

Therefore, the dimensions are 24cm x 15cm x 20cm.

Answer:

180 degrees rotation

Step-by-step explanation:

Solution:

- The congruency statements are as follows:

  Angles:      L   ≅   X

                     M   ≅   Y

                     N   ≅   Z

  Sides:        LM   ≅   XY

                    MN   ≅   YZ

                     LN    ≅   XZ  

   Transformation:

                    L ( -4 , 1 )  ------- > X ( 4 , - 1 )

                        ( x , y ) -----------> ( -x , -y )

                         180 degrees rotation

The congruence can be expressed as ∆LMN ≅ ∆XYZ. To map LMN onto XYZ, a transformation like translation, rotation, reflection, or a combination thereof can be used.

If triangles LMN and XYZ are congruent to each other, the corresponding parts must be equal in both measure and shape. The congruence statement for these triangles would be ∆LMN ≅ ∆XYZ, which implies that angle L is congruent to angle X, angle M to angle Y, and angle N to angle Z. Additionally, side LM is congruent to side XY, side MN to side YZ, and side NL to side XZ.

Regarding the transformation that maps triangle LMN onto XYZ, if they are congruent, any of the following rigid motions (also known as isometries) could be used: a translation, a rotation, a reflection, or a combination of these. Since these transformations preserve distance and angle measures, the size and shape of triangle LMN will remain unchanged, and it will match exactly onto triangle XYZ.

Yo sup??

by Pythagoras theorem we can say

38²=34²+b²

b²=1444-1156

=288

b=17

Hope this helps

Answer:

D = 17

Step-by-step explanation:

Hope this helps

Answer:

false

Step-by-step explanation:

Answer:

$0.36 per share

Step-by-step explanation:

The data provided in the question are as follows

Common stock outstanding = 50,000 shares

Preferred stock outstanding = 4,000 shares

Par value of common stock = $4

Interest rate and par value of preferred stock = 9% and $100

Total dividend payment declared = $54,000

So, the amount of dividend for each share of common stock is

= (Total dividend payment declared - Preferred stock outstanding × interest rate × par value) ÷ (common stock outstanding)

= ($54,000 - 4,000 × $100 × 9%) ÷ (50,000 shares)

= ($54,000 - $36,000) ÷ (50,000 shares)

= $18,000 ÷ 50,000 shares

= $0.36 per share

Answer:

Linda's share was $ 1933.75

Step-by-step explanation:

We first need to calculate the comission that was paid to the other real state firm wich is 7% of the sales price, so we can use the following equation:

Comission = 85000 * (7/100) = 85000*0.07 = 5950 $

Since this comission was split in half between Johnson's broker and the other broker we have 5950/2 = 2975 for each of them. From that value 65% should stay with the broker wich is Linda in this case so:

Linda's share = 2975*(65/100) = 2975*(0.65) = 1933.75 $.

Given Information:

Sale Amount = $85,000

Commission rate = 7%

Linda Johnson commission rate = 65% of 50% of commission amount

Required Information:

Linda Johnson share in the sale = ?

Answer:

Linda Johnson share in the sale = $1933.75

Step-by-step explanation:

Linda Johnson share in the sale is

Linda's share = 65% of 50% of commission amount

Where commission amount is

commission amount = 7% of sale amount

commission amount = 0.07*85,000

commission amount = $5,950

Linda's share = 0.65*0.50*5,950

Linda's share = $1933.75

Answer:

its C i think

Step-by-step explanation:

What is 0.05 Equal to 0.5.

Answer:

The 90% confidence interval is (0.383,0.497)    

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 200

Number of children that would attend  Valentine's Day Forma, x = 88

[tex]\hat{p} = \dfrac{x}{n} = \dfrac{88}{200} = 0.44[/tex]

90% Confidence interval:

[tex]\hat{p}\pm z_{stat}\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}[/tex]

[tex]z_{critical}\text{ at}~\alpha_{0.10} = \pm 1.64[/tex]

Putting the values, we get:

[tex]0.44\pm 1.64(\sqrt{\dfrac{0.44(1-0.44)}{200}}) = 0.44\pm 0.057\\\\=(0.383,0.497)[/tex]

Interpretation:

The 90% confidence interval is (0.383,0.497). We are 90% confident that the proportion of children attending Valentine's Day Formal is between 38.3% and 49.7%

Answer:

33%

Step-by-step explanation:

Assuming the weight of the mixture to be 100g**, then the weight of ryegrass in the mixture would be 30g.

Also, assume the weight mixture X used in the mixture is Xg, then the weight of mixture Y used in the mixture would be (100-X)g.

So we can now equate the parts of the ryegrass in the mixture as:

0.4X + 0.25(100-X) = 30

<=> 0.4X + 25 - 0.25X = 30

<=> 0.15X = 5

<=> X = 5/0.15 = 500/15 = 100/3

So the weight of mixture X as a percentage of the weight of the mixture

= (weight of X/weight of mixture) * 100%

= (100/3)/100 * 100%

= 33%

Answer:

Bobby around 131 minutes and Billy around 111 minutes

Step-by-step explanation:

To solve the problem it is important to raise equations regarding what happens.

 They tell us that Billy (Bi) and Bobby (Bo) can mow the lawn in 60 minutes. That is to say that what they prune in a minute is giving as follows:

1 / Bo + 1 / Bi = 1/60 (1)

They say Billy could mow the lawn only in 20 minutes less than it would take Bobby, therefore

1 / Bi = 1 / (Bo-20) (2)

Replacing (2) in (1) we have:

1 / Bo + 1 / (Bo-20) = 1/60

Resolving

(Bo - 20 + B0) / (Bo * (Bo-20) = 1/60

120 * Bo - 1200 = Bo ^ 2 - 20Bo

Rearranging:

Bo ^ 2 - 140Bo -1200 = 0

Now applying the general equation

Bo = 130.82 or Bo = 9.17, this last value cannot be because Billy took 20 minutes less and neither can he prune faster than the two together, therefore Bobby only takes around 131 minutes and Billy around 111 minutes

Checking with equation 1:

1/131 +1/111 = ~ 1/60

Final answer:

Bobby would take 120 minutes to mow the lawn by himself.

Explanation:

The question asks us to calculate how long it will take Bobby to mow the lawn by himself if it takes his twin brother Billy 20 minutes less to do the job on his own, and they can mow the lawn together in 60 minutes. To solve this, we can use the concept of work rate and the idea that the combined work rate of Billy and Bobby equals the reciprocal of the time they take to work together.

Let's assume Bobby takes x minutes to mow the lawn by himself. Thus, Billy would take x - 20 minutes. We can express their work rates as:

The combined work rate when they mow together is the sum of their individual work rates, which is equal to 1/60 since they take 60 minutes together. So, we have:

1/x + 1/(x - 20) = 1/60

By solving this equation for x, we find the time it takes Bobby to mow the lawn by himself.

Hence, Bobby will take 120 minutes to mow the lawn by himself.

Answer:

Two trapezoids labeled V R A S and B U W D. Sides B U and V R contain one tick mark. Sides B D, U W, V S, and R A each contain two tick marks. Sides A S and W D each contain three tick marks. The angles represented by vertex letters U, B, R, and V each contain two tick marks. The angles represented by vertex letters D, W, S, and A each contain one tick mark.

Answer:

b m d j and k z y a

Step-by-step explanation:

Answer:

-429/629

Step-by-step explanation:

Use angle difference formula:

sin(x − y) = sin x cos y − sin y cos x

Assuming x and y are in the first quadrant:

sin(x − y) = sin x cos y − √(1 − cos²y)√(1 − sin²x)

Plugging in values:

sin(x − y) = (8/17) (12/37) − √(1 − (12/37)²)√(1 − (8/17)²)

sin(x − y) = (8/17) (12/37) − (35/37)(15/17)

sin(x − y) = -429/629

To determine the amount of candy per sack, divide the total one fourth pound of rock candy by four, resulting in 1/16 pound of candy in each sack.

Demont has made one fourth pound of rock candy and needs to divide this equally into four sacks. To find the amount of candy in each sack, we divide the total amount of rock candy by the number of sacks.

Here's the calculation:

Total rock candy = 1/4 pound

Number of sacks = 4

Amount of candy per sack = 1/4 pound ÷ 4 = 1/16 pound per sack

Therefore, each sack will contain 1/16 pound of rock candy.

Answer:

P ( X < 4 ) = 0.3015

Step-by-step explanation:

Given:

- The ratings for current lives on a scale 0 - 10 were normally distributed with parameters mean (u) and standard deviation (s).

                                   u = 5.3

                                   s = 2.5

Find:

Find the probability that a randomly selected study​ participant's response was less than 4.

Solution:

- Declare a random variable X that follows a normally distribution with parameters u and s, mean and standard deviation respectively.

                                  X~N( 5.3 , 2.5 )

- To determine the probability of the rating to be less than 4 for a randomly selected study participant's response we have:

                                  P ( X < 4 )

- Compute the Z-score value for the limit given:

                                  P ( Z < (4 - 5.3) / 2.5 )

                                  P ( Z < -0.52 )

- Use the Z-Table to calculate the above probability as follows:

                                  P ( Z < -0.52 ) = 0.3015

- Hence, the required probability is equivalent to Z-score value probability:

                                    P ( X < 4 ) = P ( Z < -0.52 ) = 0.3015

The probability that a randomly selected study participant's response was less than 4 is 0.3015.

To find the probability that a randomly selected study participant's response was less than 4, we can use the properties of the normal distribution. Given that the mean is 5.3 and the standard deviation is 2.5, we can standardize the value 4 using the z-score formula: z = (X - µ) / σ, where X is the value of interest, µ is the mean, and σ is the standard deviation. Once we find the z-score, we look up this value in the standard normal distribution table (Z-table) or use a calculator to find the probability.

The probability that a randomly selected study participant's response was less than 4 is therefore 0.3015, or 30.15% when converted into percentage form.

Answer: the balance is $5328.95

Step-by-step explanation:

The formula for continuously compounded interest is

A = P x e (r x t)

Where

A represents the future value of the investment after t years.

P represents the present value or initial amount invested

r represents the interest rate

t represents the time in years for which the investment was made.

e is the mathematical constant approximated as 2.7183.

From the information given,

P = 3200

r = 8.5% = 8.5/100 = 0.085

t = 6 years

Therefore,

A = 3200 x 2.7183^(0.085 x 6)

A = 3200 x 2.7183^(0.51)

A = $5328.95 to the nearest hundredth

Answer:

FOR PLATO/EDMENTUM USER THE right answer is: 5328.93

Step-by-step explanation:

Answer:

Option B) 31.348  

Step-by-step explanation:

We are given the following in the question:

Sample size, n = 49

Sample standard deviation, s = 2.6 minutes

Population  standard deviation, [tex]\sigma[/tex] =  2.9 minutes

Significance level, [tex]\alpha[/tex] = 0.05

We have o find the test statistic.

Formula:

[tex]\chi^2 = \dfrac{(n-1)s^2}{\sigma^2}\\\\\chi^2 =\displaystyle\frac{(40-1)(2.6)^2}{(2.9)^2}\\\\\chi^2=31.348[/tex]

Thus, the test statistic is

Option B) 31.348

Answer:

a) Rate of change of amount

[tex]A'(t) =110e^{0.055t}[/tex]

b) &170.79

c) 0.21

Step-by-step explanation:

We are given the following in the question:

The balance is given by the equation:

[tex]A(t) =- 2000e^{0.055t}[/tex]

where t is the time in years and the initial investment is $2000 when compounded continuously.

a) Rate of change of amount

[tex]\dfrac{d(A(t))}{dt} = \dfrac{d}{dt}(2000e^{0.055t})\\\\\dfrac{d(A(t))}{dt} = 2000e^{0.055t}\times 0.055\\\\\dfrac{d(A(t))}{dt} =110e^{0.055t}[/tex]

b) We have to find the value of A'(8)

[tex]A'(t) =110e^{0.055t}\\A'(8) = 110e^{0.055(8)} = 170.79[/tex]

Interpretation:

The future value of 9 year investment of $2000 will be $170.79 more than the future value of 8 year investment.

c) Comparison

Approximation = $171

Actual change = $170.79

Difference =

[tex]\text{Approximation - Actual change}\\=171 - 170.79\\=0.21[/tex]

Thus, the error is 0.21

The formula for dA/dt is 2000(0.055)e^(0.055t). A'(8) is found by substituting t=8 into dA/dt. To compare the approximation of $171 to the actual change, subtract the initial investment from the new balance after 8 years.

a. To find the formula for dA/dt, we differentiate the equation A(t) = 2000e^(0.055t) with respect to t. Using the chain rule, we have dA/dt = 2000(0.055)e^(0.055t).

b. To find A'(8), we substitute t = 8 into the expression for dA/dt. Plugging in the values, we get A'(8) = 2000(0.055)e^(0.055(8)). This will give us the rate of change of the balance after 8 years.

c. To compare the approximation of $171 to the actual change, we subtract the initial investment of $2000 from the new balance after 8 years. The approximation is $171, so the actual change is A(8) - 2000. Plugging in the values, we get A(8) - 2000 = 2000e^(0.055(8)) - 2000.

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Answer:

-√(3/10)

Step-by-step explanation:

cos(x) = 2[cos(x/2)]² - 1

-2/5 = 2[cos(x/2)]² - 1

3/5 = 2[cos(x/2)]²

3/10 = [cos(x/2)]²

cos(x/2) = +/- sqrt(3/10)

Since x is in the 3rd quadrant, x/2 would be in the second quadrant.. so cos(x/2) is negative

[tex]x[/tex] is in quadrant III, so [tex]\pi<x<\frac{3\pi}2[/tex].

This makes [tex]\frac\pi2<\frac x2<\frac{3\pi}4[/tex], which means [tex]\frac x2[/tex] lies in quadrant II, for which we expect [tex]\cos\frac x2<0[/tex].

Recall the double-angle identity:

[tex]\cos^2\dfrac x2=\dfrac{1+\cos x}2[/tex]

[tex]\implies\cos\dfrac x2=-\sqrt{\dfrac{1-\frac25}2}=-\sqrt{\dfrac3{10}}[/tex]

Answer:

Step-by-step explanation:

Haven't seen a related rates problem in a while! These are fun!  Not too bad when you keep your stuff organized.  First, label what you've been given.  If the radius is decreasing, then we have

[tex]\frac{dr}{dt}=-.2[/tex]

We are told to find [tex]\frac{dV}{dt}[/tex] when r = 9.

Now we have to find the derivative of the volume of a sphere using implicit differentiation.  The derivative is

[tex]\frac{dV}{dt}=\frac{4}{3}\pi3r^2\frac{dr}{dt}[/tex]

It looks like we have everything we need to solve for the unknown.  The derivative is even already set up to solve for the change in volume.  All we have to do now is plug in the values.

[tex]\frac{dV}{dt}=\frac{4}{3}\pi3(81)(-.2)[/tex]

This does give us a negative number, -203.575 to be exact, but if you answer it without the negative, you say that the volume is decreasing at the rate of 203.575 cm/min cubed