Whats 4x + 5 = 6 (x + 3) - 20 - 2x

part a.

the percentage of eggs between 42 and 45mm is 48.48%

part b.

The median width is approximately (42+45)/2 = 43.5mm.

The median length is approximately (56+59)/2 = 57.5mm.

part c.

The width of grade A chicken eggs has a range of about 24mm.

part d.

I think its impossible to determine because we don't have the value for the standard deviation.

The second option should be correct.

A histogram is described as an approximate representation of the distribution of numerical data.

part a.

From the histogram, we see that the frequency for the bin that ranges from 42 to 45mm is 4 and we have  a total of 33 eggwe use this values and calculate the percentage of eggs between 42 and 45mm is 48.48%.

part b.

we have an  estimation  that the median of the width is 48mm and the median of the length is around 60mm.

part c.

Also from the histogram, we notice  that the smallest value is around 36mm and the largest value is around 66mm, hence the width of grade A chicken eggs has a range of about 24mm.

In a histogram, the range is the width that the bars cover along the x-axis and these are approximate values because histograms display bin values rather than raw data values.

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Therefore, the expression 250 * [tex]2^h[/tex] gives the number of bacteria in the colony after h hours, assuming that the initial number of bacteria is 250.

In mathematics, an expression is a combination of numbers, variables, and mathematical operations, such as addition, subtraction, multiplication, and division. Expressions can also include parentheses, brackets, and other symbols to clarify the order of operations. Expressions can be evaluated to obtain a numerical result or can be simplified using mathematical rules and properties to make them easier to work with. In algebra, expressions are often used to represent relationships between variables and to solve equations and inequalities.

Here,

In the given expression 250*[tex]2^h[/tex], the constant 250 represents the initial number of bacteria in the colony.

Since the number of bacteria doubles every hour, if we start with 250 bacteria at the beginning, after one hour we will have 250 * 2 = 500 bacteria. After two hours, we will have 500 * 2 = 1000 bacteria, and so on.

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Jasmine walks from the school to the post office, which is a distance of $2 - (-4) = 6$ blocks horizontally and 0 blocks vertically, so the distance is 6 blocks. Then she walks from the post office to the library, which is a distance of $2 - 2 = 0$ blocks horizontally and $-4 - 1 = -5$ blocks vertically, so the distance is 5 blocks.

The total distance of Jasmine's walk is the sum of the distances of each leg of her journey, which is $6 + 5 = 11$ blocks. Therefore, Jasmine walks 11 blocks in total.

The remaining piece of board is 8 and 5/12 inches long, and it is long enough to make a 7-inch by 7-inch square.

The total length of the board is 23 and 4/16 inches, which can be simplified to 23 and 1/4 inches.

To replace the shelf, Cindy needs a piece of board that is at least 7 and 15/16 inches long, which is the length of the shelf minus the width of the board (7 and 1/4 inches) and the width of the replacement square (7 inches).

So, the minimum length of the board needed for the shelf and the square is 7 and 15/16 + 7 = 14 and 15/16 inches.

Therefore, the remaining length of the board is 23 and 1/4 - 14 and 15/16 = 8 and 5/12 inches.

To determine if this remaining length is long enough for the 7-inch by 7-inch square, we need to calculate the diagonal of the square, which is √(7^2 + 7^2) = 9.899 inches (rounded to three decimal places).

Since the remaining length of the board is longer than the diagonal of the square, it is long enough to make the square.

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Answer:

Jack now makes $10.45

Step-by-step explanation:

First, we need to find 10% of 9.50. To do that we must divide 9.50 by 10 or simply move the decimal one place to the left to get 0.95. 0.95 is 10% of 9.50. Now we must ADD 0.95 to 9.50 because Jack is getting a 10% raise.  0.95 + 9.50 = 10.45. Therefore Jack makes $10.45 per hour at his job.

To determine the exact distance and bearing of the island from Miami, we can use basic trigonometry and vector addition.

First, we need to break down the flight path into its components. The initial bearing of E 20 S can be broken down into an eastward component of 600 mph [tex]cos(20°) = 562.57[/tex] mph and a southward component of 600 mph [tex]sin(20°) = 208.38[/tex] mph.

After flying for 4 hours, the plane has traveled a distance of 600 mph × 4 = 2400 miles, with a displacement of 562.57 mph × 4 = 2250.28 miles eastward and 208.38 mph × 4 = 833.52 miles southward.

When the pilot adjusts the bearing to S 5'E, the plane travels 600 mph × 2 = 1200 miles with a displacement of 600 mph cos(5°) × 2 =996.18 miles eastward and 600 mph sin(5°) × 2 = 104.57 miles southward.

Finally, when the pilot adjusts the bearing to E 5°S, the plane travels 600 mph × 1 = 600 miles with a displacement of 600 mph cos(5°) = 598.31 miles eastward and 600 mph sin(5°) = 52.42 miles southward.

To find the total displacement from Miami to the island, we can add up the eastward and southward components:

Total eastward displacement = 2250.28 + 996.18 + 598.31 = 3844.77 miles

Total southward displacement = 833.52 + 104.57 + 52.42 = 990.51 miles

Using the Pythagorean theorem, we can find the total distance from Miami to the island:

[tex]Distance = sqrt((3844.77)^2 + (990.51)^2) =3985.21 miles[/tex]

To find the bearing from Miami to the island, we can use inverse trigonometry:

[tex]Bearing = tan^{-1} (\frac{990.51}{3844.77}) = 14.76°[/tex]

Therefore, you should tell the rescue authorities in Miami that you are approximately 3985.21 miles away from Miami and the bearing to the island is approximately N 75°E.

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Answer:

[tex]x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n[/tex]

Step-by-step explanation:

Given equation:

[tex]\sin(2x)\sin(x)+\cos(x)=0[/tex]

Rewrite sin(2x) using the trigonometric identity sin(2x) = 2sin(x)cos(x):

[tex]\implies 2\sin(x)\cos(x)\sin(x)+\cos(x)=0[/tex]

[tex]\implies 2\sin^2(x)\cos(x)+\cos(x)=0[/tex]

Factor out cos(x):

[tex]\implies \cos(x)\left[2\sin^2(x)+1\right]=0[/tex]

Applying the zero-product property:

[tex]\textsf{Equation 1:}\quad\cos(x)=0[/tex]

[tex]\textsf{Equation 2:}\quad2\sin^2(x)+1=0[/tex]

Solve each part separately.

[tex]\underline{\sf Equation \; 1}[/tex]

[tex]\begin{aligned}\cos(x)&=0\\x&=\arccos(0)\\x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n\end{aligned}[/tex]

[tex]\underline{\sf Equation \; 2}[/tex]

[tex]\begin{aligned}2\sin^2(x)+1&=0\\\sin^2(x)&=-\dfrac{1}{2}\;\;\;\;\;\;\leftarrow\;\textsf{No solution}\end{aligned}[/tex]

Therefore, the solutions of the equation in radians are:

[tex]\boxed{x&=\dfrac{1}{2}\pi +2\pi n,\;\; \dfrac{3}{2}\pi + 2 \pi n}[/tex]

The number of cans that can be packed in a certain carton has correct statement as, Statements (1) and (2) together are not sufficient, option E.

Data sufficiency refers to evaluating and analysing a collection of data to see if it is sufficient to respond to a certain query. They are intended to assess the candidate's capacity to connect the dots between each question and arrive at a conclusion.

The size of each can is not revealed in statement 1 at all.

The size of the container is not disclosed in statement 2 at all.

We obtain two situations when we take into account both assertions. Case A: If the box is 1 x 1 x 2304 (inches) in size, then there are no cans that will fit within the carton.

Case B: If the box is 10 x 10 x 23.04 (inches) in size, then the carton can hold more than 0 cans.

The combined statements are insufficient because we lack clarity in our ability to respond to the target inquiry.

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Complete question:

What is the number of cans that can be packed in a certain carton?

(1) The interior volume of this carton is 2, 304 cubic inches.

(2) The exterior of each can is 6 inches high and has a diameter of 4 inches.

A. Statement (1) alone is sufficient, but statement (2) alone is not sufficient.

B. Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

C. Both statements together are sufficient, but neither statement alone is sufficient.

D. Each statement alone is sufficient.

E. Statements (1) and (2) together are not sufficient.

For -10 as the value of k the equation k – 3(k + 5) – 0. 5 = 3(0. 25k + 4) is true.

Given equation is k – 3(k + 5) – 0. 5 = 3(0. 25k + 4)

To solve the equation, firstly we have to solve or open the brackets using the distributive property that is a(x + y) = ax +ay

Thus the equation becomes, k - 3k - 15 - 0.5 = 0.75k + 12

Then we have to take all the terms with k on one side and other on the other or segregate the like terms, the equation now is

k - 3k - 0.75k = 12 + 15 + 0.5

Simplify the equation by adding or subtracting the like terms.

-2.75k = 27.5

Then we divide the coefficient of k by the other side and get our answer

k = 27.5 ÷ (-2.75)

k = -10

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The calculated value of the actual area of the playground is 5625π

From the question, we have the following parameters that can be used in our computation:

A scale drawing of a playground had a scale of 15

This means that

Scale factor = 15/1

Evaluate

Scale factor = 15

The actual area of the park in meters squared is calculated as

Area = Area of scale * Scale factor²

Substitute the known values in the above equation, so, we have the following representation

Area = Area of scale * 15²

Using the area of circle, we have

Area = π5² * 15²

Evaluate

Area = 5625π

Hence, the actual area is 5625π

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To evaluate the indefinite integral ∫x tan^2 x dx, we can use integration by parts.

Let u = x and dv = tan^2 x dx. Then, du/dx = 1 and v = ∫tan^2 x dx = tan x - x.

Using the formula for integration by parts, we have:

∫x tan^2 x dx = uv - ∫v du/dx dx
= x(tan x - x) - ∫(tan x - x) dx
= x(tan x - x) + ln|cos x| + C

Therefore, the indefinite integral of x tan^2 x dx is x(tan x - x) + ln|cos x| + C, where C is the constant of integration.
To evaluate the indefinite integral ∫x tan^2(x) dx, we can use integration by parts, which is defined as ∫udv = uv - ∫vdu.

First, let's choose our u and dv:
u = x, so du = dx
dv = tan^2(x) dx

To find v, we need to integrate dv. Since tan^2(x) = sec^2(x) - 1, we get:
v = ∫(sec^2(x) - 1) dx = tan(x) - x

Now, using integration by parts:
∫x tan^2(x) dx = x(tan(x) - x) - ∫(tan(x) - x) dx

Let's evaluate the remaining integral:
∫(tan(x) - x) dx = ∫tan(x) dx - ∫x dx
= ln|sec(x)| - (1/2)x^2 + C

So, the final answer is:
∫x tan^2(x) dx = x(tan(x) - x) - [ln|sec(x)| - (1/2)x^2] + C

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The sample regression equation for salary and education is Salaryˆ= 32.67 + 4.46Education. For each additional year of education, an individual's salary is predicted to increase by $4,460. Predicted salary for 7 years of education is $63,845.

Using the provided data, we can calculate the sample regression equation for the model  Salary = β0 + β1Education + ε by using linear regression. The result is Salaryˆ= 32.67 + 4.46Education.

The coefficient for Education is 4.46, which means that as Education increases by 1 year, an individual’s annual salary is predicted to increase by $4,460.

To find the predicted salary for an individual who completed 7 years of higher education, we substitute Education = 7 into the regression equation: Salaryˆ= 32.67 + 4.46(7) = $63,845. Therefore, the predicted salary for an individual who completed 7 years of higher education is $63,845.

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Answer:

22(15)(12) + (1/2)(22)(10)(15) = 5,610 cm^2

The measure of angle M is 20°.

To solve the problem, we need to find the measure of angle M, given the information about Circle K with center O, diameter US, angle MRU = 50°, and angle MUT = 30°.

Step 1: Determine the relationship between angles MRU and MUT.
Since MRU and MUT are both inscribed angles in Circle K, they share the same intercepted arc, which is arc MU.

Step 2: Calculate the measure of arc MU.
The measure of an intercepted arc is twice the measure of the inscribed angle. Since angle MRU = 50°, the measure of arc MU will be 2 * 50° = 100°.

Step 3: Find the measure of angle M.
We know that angle MUT = 30°, and the measure of an intercepted arc is twice the measure of the inscribed angle. Therefore, the measure of arc MT = 2 * 30° = 60°. Now, since arc MU = 100°, we can determine the measure of arc MS (arc MS = arc MU - arc MT) which is 100° - 60° = 40°.

Step 4: Calculate the measure of angle M.
Finally, the measure of angle M can be found using the intercepted arc MS. Since the measure of an intercepted arc is twice the measure of the inscribed angle, angle M = 1/2 * arc MS = 1/2 * 40° = 20°.

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Natalie will have earned a total of $488.97 in interest by March 2025.

To solve this problem, we can use the formula for compound interest:

A = [tex]P(1 + r/n)^(nt)[/tex]

where A is the total amount, 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 time in years.

First, let's find out how much money Natalie will have in her first account after 5 years:

P = $800

r = 4.8% per year = 0.048

n = 12 (compounded monthly)

t = 5 years

A = [tex]800(1 + 0.048/12)^(12*5)[/tex]

A = $995.08

So after 5 years, Natalie will have $995.08 in her first account.

Next, let's find out how much money Natalie will have in her second account:

P = $995.08

r = 6% per year = 0.06

n = 2 (compounded semiannually)

t = 5 years

A = [tex]995.08(1 + 0.06/2)^(2*5)[/tex]

A = $1,288.97

So after reinvesting her money in the second account, Natalie will have $1,288.97 after 5 years.

Finally, let's calculate the total interest earned:

Total interest = A - P

Total interest = $1,288.97 - $800

Total interest = $488.97

Therefore, Natalie will have earned a total of $488.97 in interest by March 2025.

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We want to use properties to write expressions for the length of the other sides of the square.

Remember that the length of all the sides in a square is the same, so we only need to rewrite the above expression in two different ways.

First, we can use the distribute property in the first term:

[tex]\sf 6\times(3x + 8) + 32 + 12\times x[/tex]

[tex]\sf = 6\times3x + 6\times8 + 32 +12\times x[/tex]

[tex]= \sf 18\times x + 48 + 32 + 12\times x[/tex]

So this can be the length of one of the sides.

Now we can keep simplifying the above equation:

[tex]= \sf 18\times x + 48 + 32 + 12\times x[/tex]

To do it, we can use the distributive and associative property in the next way:

[tex]\sf 18\times x + 48 + 32 + 12\times x[/tex]

[tex]= \sf 18\times x + 12\times x + 48 + 32[/tex]

[tex]= \sf (18\times x + 12\times x) + (48 + 32)[/tex]

[tex]= \sf (18 + 12)\times x + 80[/tex]

[tex]= \sf 30\times x + 80[/tex]

This can be the expression to the other side.

The value of measure of angle WPN is,

⇒ m ∠WPN = 108°

We have to given that;

Parallel lines EN, BH, and RK, with transversal PW are shown.

And, m ∠ BMV = 108° and m ∠KVS = 72°

Now, We get by definition of vertically opposite angle;

m ∠ BMV = m ∠PWN = 108°

Hence, By definition of alternate angle we get;

⇒ m ∠PWN = m ∠WPN = 108°

Thus, The value of measure of angle WPN is,

⇒ m ∠WPN = 108°

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As per the probability, the expected demand for bananas per week is 182.5 pounds.

To model this problem, we can use decision analysis, which involves identifying the possible outcomes, assigning probabilities to each outcome, and calculating the expected value of each decision.

In this case, the possible outcomes are the demand for bananas, which can be 100, 150, 200, or 250 pounds per week. The probabilities of each demand level are given as 0.20, 0.25, 0.35, and 0.20, respectively.

Let X denote the demand for bananas in pounds. Then, the expected demand for bananas, denoted as E(X), can be calculated as follows:

E(X) = 100(0.20) + 150(0.25) + 200(0.35) + 250(0.20) = 182.5

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The runner's average speed is approximately 11.63 mph, which is slightly faster than 11 mph. So, the runner doesn't maintain exactly 11 mph throughout the entire race, but they are close.

If a runner completes a 6.2 mile race in 32 minutes, we can calculate their average speed by dividing the total distance by the total time.

6.2 miles ÷ 32 minutes = 0.194 miles per minute

To convert this to miles per hour, we need to multiply by 60 (the number of minutes in an hour):

0.194 miles per minute x 60 minutes = 11.63 miles per hour

So the runner must be running at an average speed of 11.64 miles per hour throughout the entire race in order to complete it in 32 minutes. This is an impressive pace and shows that the runner is very fit and capable of sustaining a fast speed for a relatively long distance.

A runner completes a 6.2-mile race in 32 minutes, and we are to determine if they maintain an 11 mph pace throughout the race. To do this, we need to convert the race time to hours and then divide the race distance by the time.

First, convert 32 minutes to hours:
32 minutes / 60 minutes/hour = 0.5333 hours

Next, calculate the average speed:
6.2 miles / 0.5333 hours ≈ 11.63 mph

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There are 15,660 different ways a committee consisting of 2 teachers and 2 students can be formed from 9 teachers and 30 students.

To find out how many different committees can be formed from 9 teachers and 30 students, if the committee consists of 2 teachers and 2 students, we will use the combination formula. The combination formula is given by C(n, r) = n! / (r!(n-r)!), where n is the total number of items and r is the number of items to be selected.

First, let's find the number of ways to select 2 teachers from 9:
C(9, 2) = 9! / (2!(9-2)!) = 9! / (2! * 7!) = 36

Next, let's find the number of ways to select 2 students from 30:
C(30, 2) = 30! / (2!(30-2)!) = 30! / (2! * 28!) = 435

Now, to find the total number of ways the committee of 4 members can be selected, we simply multiply the number of ways to select teachers and students:
Total ways = 36 (ways to select teachers) * 435 (ways to select students) = 15,660

So, there are 15,660 different ways a committee consisting of 2 teachers and 2 students can be formed from 9 teachers and 30 students.

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The inequality to describe the number of subscriptions Andre must sell to reach his goal is  3s + 25 ≥ C.

What are inequalities ?

Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.

here , we have,

The symbol a < b indicates that a is smaller than b.

When a > b is used, it indicates that a is bigger than b.

a is less than or equal to b when a notation like a ≤ b.

a is bigger or equal value of an is indicated by the notation a ≥ b.

Assuming the cost of soccer cleats to be 'C' and the number of subscriptions to be 's'.

∴ The inequality that represents this situation is 3s + 25 ≥ C to real his goal.

Hence, The inequality to describe the number of subscriptions Andre must sell to reach his goal is  3s + 25 ≥ C.

Answer: $532,000

Step-by-step explanation:

If the company is making $140,000 and they get 20% each year, we just multiply it by 20%, or 0.20, and get $28,000. So, we would multiply that by 14, the years the company operated, and then add it to the original $140,000.

28,000 x 14 = 392,000

392,000 + 140,000 = 532,000

So, over the course of 14 years, the company made a profit of $532,000.

Answer:

C)4.192

Step-by-step explanation:

What is 524/125 written as a decimal?

We Take

524 divided by 125 = 4.192

So, the answer is C)4.192

The average price of a gallon of milk in the following years, using the exponential growth function, are:

a) 2018 = $2.90

  2021 = $3.55

b) Based on the exponential growth function, the cost of milk is inflating at 7% per year.

c) Based on the percentage of inflation, the predicted price of a gallon of milk in 2025 is $4.66.

An exponential growth function is a mathematical equation that describes the relationship between two variables (dependent and independent).

Under the function, there is a constant ratio of growth with the number of years between the initial value and the desired value as the exponent.

The given function for the price of average gallon of milk from 2008 to 2021 is 3.55 = 2.90 (1 + x)³.

Average price of milk in 2018 = $2.90

Average price of milk in 2021 = $3.55

Change in the average price of milk = $0.65 ($3.55 - $2.90)

The percentage change from 2018 to 2021 = 22.41% ($0.65 ÷ $2.90 x 100)

The cost of milk is inflating annually at (1 + x)^3

x = 7%

Cost of milk in 2025 = y

Number of years from 2018 to 2025 = 7 years

y = 2.90 (1 + 0.07)⁷

y = 2.90 (1.07)⁷

y = $4.66

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a) The number of cities that had only a professional sports team can be found by subtracting the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had all three activities from the total number of cities:

33 - (9 + 6 + 3) = 15 cities had only a professional sports team.

b) The number of cities that had a professional sports team and a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the number of cities that had a professional sports team and a symphony:

9 - 3 = 6 cities had a professional sports team and a symphony, but not a children's museum.

c) The number of cities that had a professional sports team or a symphony can be found by adding the number of cities that had a professional sports team, the number of cities that had a symphony, and then subtracting the number of cities that had both:

17 + 15 - 9 + 14 - 6 + 3 = 34 cities had a professional sports team or a symphony.

d) The number of cities that had a professional sports team or a symphony, but not a children's museum can be found by subtracting the number of cities that had all three activities from the answer to part c:

34 - 3 = 31 cities had a professional sports team or a symphony, but not a children's museum.

e) The number of cities that had exactly two of the activities can be found by adding up the number of cities that had a professional sports team and a symphony, the number of cities that had a professional sports team and a children's museum, and the number of cities that had a symphony and a children's museum, and then subtracting twice the number of cities that had all three activities:

9 + 6 + 6 - 2(3) = 15 cities had exactly two of the activities.

Using the change of variables u = x and v = x + y, we transform the given region R into a rectangle S, and evaluate the integral as 9 (e^6 - 2e^4 + e^2 - 1).

We need to find a change of variables that maps the region R onto a rectangle in the uv-plane. Let's make the following substitutions

u = x

v = x + y

Then, the region R is transformed into the rectangle S defined by 0 ≤ u ≤ 1 and 0 ≤ v ≤ 2.

To find the limits of integration in the new variables, we can solve the equations 2[x] + 2y = 2 for x and y in terms of u and v

2[x] + 2y = 2

2u + 2v - 2[x] = 2

[x] = u + v - 1

Since [x] is the greatest integer less than or equal to x, we have

u + v - 1 ≤ x < u + v

Also, since 0 ≤ y ≤ 1, we have

0 ≤ x + y - u ≤ 1

u ≤ x + y < u + 1

u - x ≤ y < 1 + u - x

Now we can evaluate the integral using the new variables

∫∫R 9e^(2x+2y) dA = ∫∫S 9e^(2u+2v) |J| dudv

where J is the Jacobian of the transformation, given by

|J| = det [[∂x/∂u, ∂x/∂v], [∂y/∂u, ∂y/∂v]]

= det [[1, 1], [-1, 1]]

= 2

Therefore, the integral becomes

∫∫S 9e^(2u+2v) |J| dudv = 2 ∫0^1 ∫0^2 9e^(2u+2v) dudv

= 2 ∫0^1 [9e^(2u+2v)/2]_0^2 dv

= 2 ∫0^1 (9/2)(e^(4+2v) - e^(2v)) dv

= 2 (9/2) [(e^6 - e^2)/2 - (e^4 - 1)/2]

= 9 (e^6 - 2e^4 + e^2 - 1)

Therefore, the value of the integral is 9 (e^6 - 2e^4 + e^2 - 1).

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The possible values of Y are 0, 1, 2, 3, and 4.

The number of female children in a family with four children can be any value between 0 and 4, inclusive.

To see why, we can consider all the possible outcomes of the family having four children, assuming that the probability of having a boy or a girl is 0.5 (assuming a binomial distribution).

There are 2 possibilities for the first child (boy or girl), 2 possibilities for the second child, 2 possibilities for the third child, and 2 possibilities for the fourth child, making a total of 2x2x2x2 = 16 possible outcomes.

Out of these 16 outcomes, we can count the number of outcomes that correspond to each possible value of Y:

If Y = 0, then all four children must be boys, which is 1 outcome.

If Y = 1, then there are 4 ways to have one girl (first, second, third, or fourth child).

If Y = 2, then there are 6 ways to have two girls (first two, first three, first four, second three, second four, or third fourth child).

If Y = 3, then there are 4 ways to have three girls (first three, first four, second four, or third four child).

If Y = 4, then all four children must be girls, which is 1 outcome.

Therefore, the possible values of Y are 0, 1, 2, 3, and 4.

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We know that the length of TU to the nearest tenth is 6.9 in

The figure is tangent to the circle at point U. Using the information given in the figure, we can conclude that segment ST is tangent to the circle at point T.

To find the length of TU, we can use the formula for the length of a tangent segment from a point outside the circle:

TU^2 = TS x TR

We know that TS = ST = 4 in. To find TR, we can use the Pythagorean theorem:

TR^2 = RS^2 - TS^2
TR^2 = 8^2 - 4^2
TR^2 = 48
TR = sqrt(48)

Now we can substitute the values we have found into the first formula:

TU^2 = 4 x sqrt(48)
TU = sqrt(4 x sqrt(48))
TU ≈ 6.9 in.

Therefore, the length of TU to the nearest tenth is 6.9 in.

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Step-by-step explanation:

(10 x 6)/2 = 30 + 47 = 77

To find the average temperature of the coffee during the first half hour, we need to find the temperature of the coffee at t = 0 (when the coffee is just brewed) and at t = 30 (after half an hour has passed).

At t = 0, T(0) = 20 + 75e^0/50 = 20 + 75 = 95 C. At t = 30, T(30) = 20 + 75e^-30/50 ≈ 42.5 C.

So, the temperature of the coffee decreases from 95 C to 42.5 C during the first half hour.

The average temperature during this time period can be found by taking the average of the initial and final temperatures:

Average temperature = (95 C + 42.5 C) / 2 = 68.75 C.

Therefore, the average temperature of the coffee during the first half hour is 68.75 C.

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