What is the value of x log3 x=4

Answer: 16

Step-by-step explanation: to find diameter its radius x 2, so 2 x 8 is 16

[tex]8 \frac{1}{3} \pi[/tex]

Step-by-step explanation:

volume of a sphere = 4/3 pi r²

r = 2.5

4/3× pi× 2.5² = 25/3pi

25/3 as a mixed number is 8 and 1/3

therefore rhe answer is 8 and 1/3 pi

Answer:

y=x+8

Step-by-step explanation:

since you are starting with the linear function y=x+7

a translation one unit to the left would be y=x+7(+1)

which gives us the answer y=x+8

Answer:

(-13)^3

Step-by-step explanation:

Exponents can be used for repeated multiplication.

In this case, the number "negative 13" is repeated several times, all connected with multiplication.

There are a total of three "negative 13"s being multiplied together ("negative 13" appears three times on the page).

To rewrite using exponents, we would write one of the following:

(-13)^3

[tex](-13)^3[/tex]

Chris cannot build the triangular raised bed frame with the given lumber.

To determine if Chris can build her triangular raised bed frame, we need to check if the length of any one of the lumber pieces is greater than the sum of the other two. If this condition is not met, the pieces can be used to build the frame.

Let's check:

4 + 5 = 9 (no)

4 + 9 = 13 (no)

5 + 9 = 14 (yes)

Since the length of the 5-foot and 9-foot lumber pieces add up to be greater than the 4-foot piece, Chris can build her triangular raised bed frame. She can use the 4-foot and 5-foot pieces for the two shorter sides of the triangle and the 9-foot piece for the longer side.

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The value of the double integral is zero.

The order of integration is dr dy, which means we first integrate with respect to r and then with respect to y.

Thus, we can write the integral as:

[tex]\int^0_{2\pi} \int^0_{1 + cos(a)}[/tex] r sin(θ) dr dy

Here, we have used the given limits of integration for r and y. Now, we integrate with respect to r first, treating y as a constant.

∫r sin(θ) dr = -cos(θ)r

We can substitute the limits of integration for r, which gives:

-cos(θ)(1+cos(a)) + cos(θ)(0)

Simplifying this expression, we get:

-cos(θ)(1+cos(a))

Now, we integrate this expression with respect to y, using the limits 0 to 2π for θ.

[tex]\int ^0_{2\pi}[/tex] -cos(θ)(1+cos(a)) dy

We can integrate this expression by treating cos(a) as a constant and using the formula for integrating cosine functions:

Integral of cos(x) dx = sin(x) + C

Thus, we have:

(1+cos(a)) Integral from 0 to 2π of cos(θ) dy

= - (1+cos(a)) [sin(2π) - sin(0)]

= 0

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a. d, days since first measurement y, acres burned since fire started

0 0, 1 4800, 2 9600, 3 14400, 4 19200, 5 24000. b. when d = -1, it tells that the area burned in the fire was 4780 acres one day before the area was first measured. When d = -3, y = 4680, this means that the area burned in the fire was 4680 acres three days before the area was first measured. c. The area burned a week before the fire measured 4800 acres was approximately 115.2 acres.

a. To complete the table, we need to plug in the values of d in the given equation and calculate the corresponding values of y.

d, days since first measurement

y, acres burned since fire started

0 0

1 4800

2 9600

3 14400

4 19200

5 24000

b. When d = -1, we have:

y = (4800)(24^(-1))^(1) = 4800 - 20 = 4780

This means that the area burned in the fire was 4780 acres one day before the area was first measured.

When d = -3, we have:

y = (4800)(24^(-3))^(1) = 4800 - 120 = 4680

This means that the area burned in the fire was 4680 acres three days before the area was first measured.

c. A week before the fire measured 4800 acres, the number of days since the fire started would be:

d = 4800 / (4800/24) = 24

Therefore, a week before the fire measured 4800 acres, the fire had been burning for 24 days. Plugging in this value in the given equation, we get:

y = (4800)(24^(-24/24))^(1) = 115.2

So, the area burned a week before the fire measured 4800 acres was approximately 115.2 acres.

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The given series are in conditionally convergent

To determine whether the given series is absolutely convergent, conditionally convergent, or divergent, we will use the Comparison Test.
Series in question:
∑ [[tex]5(k(-1)^k + 1)] / (k^2),[/tex] k = 1 to ∞
Step 1: Find the absolute value of the series
| 5([tex]k(-1)^k + 1) / k^2[/tex] |
Step 2: Simplify the absolute value
[tex]5(k + (-1)^k) / k^2[/tex]
Step 3: Use the Comparison Test
We will compare this series to the series ∑ 5k / [tex]k^2,[/tex] k = 1 to ∞.

Since [tex](-1)^k[/tex] is always either 1 or -1, we know that [tex]5(k + (-1)^k) / k^2 \leq 5k / k^2.[/tex]
Step 4: Determine if the comparison series converges
The comparison series can be simplified as

∑ 5 / k, k = 1 to ∞, which is a harmonic series that is known to be divergent.
Step 5: Determine the original series' convergence status
Since the comparison series is divergent, we cannot determine if the original series is absolutely convergent using the Comparison Test.
However, we can now investigate if the series is conditionally convergent by considering the alternating series

∑ (-1)^k(5k) / [tex]k^2[/tex], k = 1 to ∞.
Since the series' terms decrease in magnitude (5k / [tex]k^2[/tex] decreases as k increases) and the limit of the terms as k approaches infinity is zero, the series is conditionally convergent by the Alternating Series Test.
In conclusion, the given series is conditionally convergent.

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The 7/8 and 5/6, when renamed using a common denominator of 24, become 21/24 and 20/24

The two fractions 7/8 and 5/6 need to be renamed using a common denominator.

To find the common denominator, we must first identify a common multiple of the denominators 8 and 6.

The smallest common multiple of 8 and 6 is 24. We can convert both fractions to have a denominator of 24 by multiplying the numerator and denominator of 7/8 by 3/3 and the numerator and denominator of 5/6 by 4/4.

This results in 21/24 and 20/24, respectively.  

Renaming fractions with a common denominator allows us to compare them more easily or perform arithmetic operations on them, which is essential in mathematical problem-solving.

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The calculated height of the mountain is approximately 548.5 ft

We can use trigonometry to solve this problem. Let h be the height of the mountain. Then we have:

sin(47°) = h / 750

Multiplying both sides by 750, we get:

h = 750 sin 47°

Using a calculator, we find:

h ≈ 548.5 ft

Therefore, the height of the mountain is approximately 548.5 ft

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Considering the definition of probability, the probability that a randomly selected attraction at this amusement park would be a drink stand is 1/10.

Probability establishes a relationship between the number of favorable events and the total number of possible events.

The probability of any event A is defined as the ratio between the number of favorable cases (number of cases in which event A may or may not occur) and the total number of possible cases:

P(A)= number of favorable cases÷ number of possible cases

In this case, you know:

Replacing in the definition of probability:

P(A)= 2÷ 20

Solving:

P(A)= 1/10

Finally, the probability in this case is 1/10.

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To find the volume of the solid above the triangle R and below the plane z = x, we can use a triple integral with cylindrical coordinates.

First, we can note that the triangle R lies in the x-y plane and is symmetric with respect to the line y = x. Therefore, we can consider the solid above the portion of R in the first quadrant and then multiply the result by 4 to get the total volume.

In cylindrical coordinates, we have:

z = r cos(theta)

x = r sin(theta)

The bounds for r and theta can be obtained by considering the equations of the lines that bound the portion of R in the first quadrant. These lines are:

y = (1/2) x

y = 4 - (1/2) x

Solving for x and y in terms of r and theta, we get:

x = r sin(theta)

y = r cos(theta)

Substituting these expressions into the equations of the lines and solving for r, we get:

r = 8 sin(theta) / (3 + 2 cos(theta))

The bounds for theta are 0 and pi/2, since we are considering the portion of R in the first quadrant.

The bounds for z are from z = 0 to z = x = r sin(theta).

Therefore, the triple integral for the volume is:

V = 4 * ∫[0, pi/2] ∫[0, 8 sin(theta) / (3 + 2 cos(theta))] ∫[0, r sin(theta)] 1 dz dr dtheta

This integral can be evaluated using standard techniques, such as trigonometric substitution. The result is:

V = 32/3

Therefore, q5 = 32/3.

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Claire's bank balance being -45.32 euros last month meant that she had spent more money than she had in her account, resulting in an overdraft. This means that she owed the bank money and would have to pay back the amount she had spent beyond her account balance along with any associated fees.

However, this month, Claire's bank balance has increased to 17.92 euros. This means that she has deposited money into her account or received a payment that has increased her account balance. It could also mean that she has spent less money than she has earned, resulting in a positive balance.

Having a positive bank balance is always a good thing because it means that you have money to spend and you are not in debt. It is important to keep track of your bank balance regularly and make sure that you do not overspend beyond your means to avoid overdraft fees and financial difficulties.

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a) The diameter d of the wheel has bounds:

35 cm ≤ d ≤ 45 cm

b) The circumference C has bounds, using C = πd:

π * 35 cm ≤ C ≤ π * 45 cm

The inequality representing the lower and upper bounds for the diameter d is:

35 cm ≤ d ≤ 45 cm

b) For the lower bound, we substitute the lower bound of the diameter (35 cm) into the formula:

[tex]C_l_o_w_e_r[/tex] = π * 35 cm

For the upper bound, we substitute the upper bound of the diameter (45 cm) into the formula:

[tex]C_u_p_p_e_r[/tex] = π * 45 cm

The inequality representing the lower and upper bounds for the circumference C is:

π * 35 cm ≤ C ≤ π * 45 cm

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The amount in pounds of kale Lori bought is 1 1/2 pounds.

To find out how many pounds of kale Lori bought, you need to multiply the total weight of the vegetables by the fraction that represents the proportion of kale.

In this case, you can calculate the amount of kale as follows:

(7 1/2) * (1/5)

First, convert the mixed number 7 1/2 to an improper fraction:

(7 * 2 + 1) / 2 = 15/2

Now multiply the two fractions:

(15/2) * (1/5)

Multiply the numerators together and the denominators together:

(15 * 1) / (2 * 5) = 15 / 10

Now, simplify the fraction:

15 ÷ 5 / 10 ÷ 5 = 3 / 2

So, Lori bought 3/2 or 1 1/2 pounds of kale from the grocery store. This means that out of the total 7 1/2 pounds of vegetables she purchased, 1 1/2 pounds were kale.

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The average rate of change for the function f(x) = x² + 6x between x = 0 and x = 7 is 13.

To find the average rate of change for the given function f(x) = x² + 6x between x = 0 and x = 7, you can follow these steps: Calculate the value of the function at the given points.
f(0) = 0² + 6(0) = 0
f(7) = 7² + 6(7) = 49 + 42 = 91

Use the average rate of change formula, which is (f(b) - f(a)) / (b - a), where a and b are the given points.

Substitute the values into the formula:
Average rate of change = (f(7) - f(0)) / (7 - 0) = (91 - 0) / 7 = 91 / 7 = 13

Therefore the average rate of change for the function f(x) = x² + 6x between x = 0 and x = 7 is calculated to be 13.

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By accurately listing your possessions and their values, you can ensure that you will receive the appropriate amount of money in case any of your belongings are lost or stolen.

To create this list, consider the following steps:
List your possessions: Start by creating a list of all the items you currently own and the ones you plan to purchase before moving into the apartment. Include furniture, electronics, appliances, clothing, and other personal belongings.
Determine the value of each item: Next, assign a monetary value to each item on your list. This can be the amount you paid for the item, its current market value, or a rough estimate based on similar items in the market.
Add up the total value: Add up the monetary values of all the items on your list to get the total value of your possessions. This will help you determine the amount of renters' insurance coverage you need to protect your belongings.
Research renters' insurance policies: Look for insurance providers that offer renters' insurance policies and compare their coverage options, deductibles, and premiums. Choose a policy that best suits your needs and budget.
Provide the list to the insurance provider: Once you have chosen an insurance provider, submit your list of possessions and their values. This will help them determine the appropriate coverage amount and premium for your renters' insurance policy.
Remember, by accurately listing your possessions and their values, you can ensure that you will receive the appropriate amount of money in case any of your belongings are lost or stolen.

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Chris must work at least 36 hours to earn a minimum of $300, assuming she sells 15 books and earns the $2 bonus for each book sold.

The inequality that represents the situation is: 7.50h + 2(15) ≥ 300 where "h" represents the number of hours Chris works, and "2(15)" represents the bonus earned for selling 15 books.

The left-hand side of the inequality calculates Chris's total earnings, which is the product of her hourly wage of $7.50 and the number of hours worked, plus the bonus earned for selling 15 books.

The inequality states that the total earnings must be greater than or equal to $300, which is the minimum amount Chris wants to earn. To solve the inequality, we can simplify it by first multiplying 2 and 15 to get 30: 7.50h + 30 ≥ 300

We can isolate "h" by subtracting 30 from both sides: 7.50h ≥ 270. we can solve for "h" by dividing both sides by 7.50: h ≥ 36.

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Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are -  A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).

In geometry, a dilation is a transformation that alters an object's size without altering its general shape.

Dilation is a particular kind of transformation in geometry that modifies an object's size while maintaining its overall shape.

Given:

scale factor = 1/2

coordinates of Quadrilateral ABCD

A(3,4) , B(1,10) ,C(12,14), D(9,3)

Now, coordinates about the dilation with centre C:, multiply each coordinate with 1/2.

A'(3*1/2,4*1/2)   --> A'(1.5, 2)

B'(1*1/2,10*1/2)    ---> B'(0.5, 5)

C'(12*1/2,14*1/2), --> C'(6, 7)

D'(9*1/2,3*1/2) ---> D'(4.5, 1.4)

Thus, the coordinates of Quadrilateral A'B'C'D' after the dilation with the scale factor of 1/2 are -  A'(1.5, 2), B'(0.5, 5), C'(6, 7), D'(4.5, 1.4).

Graph is attached.

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The y-values that a function approaches when the x-values are extremely large or extremely small is called the function's asymptotic behavior.

When we talk about the asymptotic behavior of a function, we are referring to what happens to the values of the function as the input (x-values) either tends to positive infinity or negative infinity.

In other words, we are interested in how the function behaves when the input values become extremely large or extremely small.

To understand asymptotic behavior, let's consider two types of asymptotes: horizontal and vertical asymptotes.

Horizontal Asymptotes:

A horizontal asymptote is a horizontal line that a function approaches as the x-values become extremely large or extremely small. We usually denote horizontal asymptotes as y = c, where c is a constant.

For example, let's consider the function f(x) = (2x^2 + 3) / (x^2 - 1). As x approaches positive or negative infinity, we can observe the following behavior:

As x becomes extremely large or extremely small, the function becomes closer and closer to the line y = 2. Therefore, we say that y = 2 is a horizontal asymptote for this function.

Vertical Asymptotes:

A vertical asymptote is a vertical line that the function approaches as the x-values approach a particular value. It typically occurs when there is a division by zero or when the function tends to infinity at a specific point.

For example, consider the function g(x) = 1 / (x - 2). As x approaches 2 from either side (but never equal to 2), we can observe the following behavior:

As x approaches 2 from the left (x < 2), the function g(x) becomes increasingly negative, tending towards negative infinity.

As x approaches 2 from the right (x > 2), the function g(x) becomes increasingly positive, tending towards positive infinity.

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The volume of the solid is (7π - 8√2)/12 cubic units.

To find the volume of the solid enclosed by the cone and sphere in cylindrical coordinates, we first need to express the equations of the cone and sphere in cylindrical coordinates.

Cylindrical coordinates are expressed as (ρ, θ, z), where ρ is the distance from the origin to a point in the xy-plane, θ is the angle between the x-axis and a line connecting the origin to the point in the xy-plane, and z is the height above the xy-plane.

The cone z = x^2 + y^2 can be expressed in cylindrical coordinates as ρ^2 = z, and the sphere x^2 + y^2 + z^2 = 2 can be expressed as ρ^2 + z^2 = 2.

To find the limits of integration for ρ, θ, and z, we need to visualize the solid. The cone intersects the sphere at a circle in the xy-plane with radius 1. We can integrate over this circle by setting ρ = 1 and integrating over θ from 0 to 2π.

The limits of integration for z are from the cone to the sphere. At ρ = 1, the cone and sphere intersect at z = 1, so we integrate z from 0 to 1.

Therefore, the volume of the solid enclosed by the cone and sphere in cylindrical coordinates is

V = ∫∫∫ ρ dz dρ dθ, where the limits of integration are

0 ≤ θ ≤ 2π

0 ≤ ρ ≤ 1

0 ≤ z ≤ ρ^2 for ρ^2 ≤ 1, and 0 ≤ z ≤ √(2 - ρ^2) for ρ^2 > 1.

Integrating over z, we get

V = ∫∫ ρ(ρ^2) dρ dθ for ρ^2 ≤ 1, and

V = ∫∫ ρ(√(2 - ρ^2))^2 dρ dθ for ρ^2 > 1.

Evaluating the integrals, we get

V = ∫0^1 ∫0^2π ρ^3 dθ dρ = π/4

and

V = ∫1^√2 ∫0^2π ρ(2 - ρ^2) dθ dρ = π/3 - 2√2/3

Therefore, the total volume of the solid enclosed by the cone and sphere in cylindrical coordinates is

V = π/4 + π/3 - 2√2/3

= (7π - 8√2)/12 cubic units

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The given question is incomplete, the complete question is:

Use cylindrical coordinates Find the volume of the solid that is enclosed by the cone z = x^2 + y^2 and the sphere x^2 + y^2 + z^2 = 2

Answer:

This function is an exponential function because the base is a constant and the exponent is a variable.

The sum of money shared by both Aziz and Ahmad is $80.

To find the sum of money shared by Aziz and Ahmad, we'll use the given ratio and the difference between their shares.

1. We are given that Aziz and Ahmad share the money in the ratio 3:7. Let's represent Aziz's share as 3x and Ahmad's share as 7x.
2. It's mentioned that Aziz received $32 less than Ahmad. So, we can write an equation as follows: 7x - 3x = $32.
3. Simplify the equation: 4x = $32.
4. Solve for x: x = $32 / 4, x = $8.
5. Now, we can find the shares of Aziz and Ahmad. Aziz's share: 3x = 3 * $8 = $24. Ahmad's share: 7x = 7 * $8 = $56.
6. To find the total sum of money shared, add both shares: $24 + $56 = $80.

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In this context, the slope of the line of best fit, represented by the equation y = 1.25x + 7, represents the relationship between the age of an infant (in months) and its weight (in pounds) and the independent variable, x, represents the age of the infant in months, and the dependent variable, y, represents the weight of the infant in pounds.

The slope of the line, 1.25, indicates the rate at which the infant's weight changes with respect to its age. Specifically, it shows that for each additional month of age, the infant's weight is expected to increase by 1.25 pounds. This means that, on average, an infant gains 1.25 pounds per month.

In conclusion, the slope (1.25) in this context represents the average weight gain per month for an infant, based on the data collected by the doctor. It helps to understand the general association between an infant's age and its weight, and can be useful in predicting an infant's weight at a given age. However, it's important to remember that this is an average value and individual infants may have different weight gain patterns.

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The angle of elevation of the room is approximately 19.1 degrees.

To determine the angle of elevation of the room, we need to use trigonometry. The angle of elevation is the angle between the horizontal plane and the line of sight from the rover to the room. We can use the tangent function to find this angle:

tan(angle of elevation) = opposite/adjacent

In this case, the opposite side is the vertical distance of 65 meters and the adjacent side is the horizontal distance of 180 meters. So we have:

tan(angle of elevation) = 65/180

To find the angle of elevation, we can take the inverse tangent (or arctan) of both sides:

angle of elevation = arctan(65/180)

Using a calculator, we get:

angle of elevation = 19.1 degrees (rounded to the nearest thousands of degree)

Therefore, the angle of elevation of the room is approximately 19.1 degrees.

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The total surface area of the figure shown as a rectangular prism is 208. 6 cm

The formula for calculating the total surface area of a rectangular prism is expressed as;

TSA =  2 (lh +wh + lw )

Such that the parameters are;

Now, substitute the values, we have;

TSA = 2(2(9) + 7.9(9) + 2(7.6)

expand the bracket, we have;

TSA = 2(18 + 71.1 + 15.2)

add the values

TSA = 2(104. 3)

TSA = 208. 6

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Answer:the two sides still equal.

Step-by-step explanation:To solve 2x + 3 = 5, Sylvia first subtracted 5 from both sides. What did she do wrong? Because Sylvia did the same operation to both sides, the equation is still correct: the two sides still equal.

Answer:

1

Step-by-step explanation:

or, 2x=5-3

or, 2x=2

or, x=2/2

x=1

37.5% of the students collected between 49 and 98 kilograms of newspapers for the recycling project. Percentage method can be used here.

To answer this question, we need to determine the number of students who collected between 49 and 98 kilograms of newspapers and then calculate what percentage of the total number of students that represents.

First, we need to gather the data and sort it into categories. We can create a frequency table with intervals of 10 kilograms:
Mass Range | Number of Students
   0-9 kg               3
   10-19 kg            5
  20-29 kg           7
  30-39 kg           4
  40-49 kg           6
  50-59 kg           8
  60-69 kg           5
  70-79 kg           2
  80-89 kg           1
  90-99 kg           2

To find the number of students who collected between 49 and 98 kilograms of newspapers, we need to add up the frequencies for the 50-59, 60-69, and 70-79 kg categories. That gives us a total of 15 students.
To calculate the percentage of students who collected between 49 and 98 kilograms of newspapers, we need to divide the number of students in that range by the total number of students and then multiply by 100. In this case, we have 15 students in the range and a total of 40 students overall, so:
15/40 * 100 = 37.5%.

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Answer: (a) To find the total number of fish that enter the lake over the 5-hour period from midnight to 5 A.M., we need to integrate the rate of fish entering the lake over this time period:

Total number of fish = ∫0^5 E(t) dt

Using the given function for E(t), we get:

Total number of fish = ∫0^5 (20 + 15 sin(πt/6)) dt

Using integration rules, we can solve this:

Total number of fish = 20t - (90/π) cos(πt/6) | from 0 to 5

Total number of fish = (100 - (90/π) cos(5π/6)) - (0 - (90/π) cos(0))

Total number of fish ≈ 121

Therefore, approximately 121 fish enter the lake over the 5-hour period.

(b) To find the average number of fish that leave the lake per hour over the 5-hour period, we need to calculate the total number of fish that leave the lake over this time period and divide by 5:

Total number of fish leaving the lake = L(0) + L(1) + L(2) + L(3) + L(4) + L(5)

Total number of fish leaving the lake = (4 + 20.1(0)^2) + (4 + 20.1(1)^2) + (4 + 20.1(2)^2) + (4 + 20.1(3)^2) + (4 + 20.1(4)^2) + (4 + 20.1(5)^2)

Total number of fish leaving the lake ≈ 257.5

Average number of fish leaving the lake per hour = Total number of fish leaving the lake / 5

Average number of fish leaving the lake per hour ≈ 51.5

Therefore, approximately 51.5 fish leave the lake per hour on average over the 5-hour period.

(c) To find the time when the greatest number of fish are in the lake, we need to find the maximum value of the function N(t) = E(t) - L(t) over the interval 0 ≤ t ≤ 8. We can do this by taking the derivative of N(t) with respect to t and setting it equal to zero:

N'(t) = E'(t) - L'(t)

N'(t) = (15π/6)cos(πt/6) - 40.2t

Setting N'(t) = 0, we get:

(15π/6)cos(πt/6) - 40.2t = 0

Simplifying and solving for t gives:

t ≈ 2.78 or t ≈ 6.22

Since 0 ≤ t ≤ 8, the time when the greatest number of fish are in the lake is t ≈ 2.78 hours after midnight (approximately 2:47 A.M.) or t ≈ 6.22 hours after midnight (approximately 6:13 A.M.).

To justify this, we can use the second derivative test. Taking the second derivative of N(t) gives:

N''(t) = -(15π2/36)sin(πt/6) - 40.2

At t ≈ 2.78, N''(t) is negative, which means that N(t) has a local maximum at this point. Similarly, at t ≈ 6.22, N''(t) is positive, which also means that N(t) has a local maximum at this point. Therefore, these are the times when the greatest number of fish are in the lake.

(d) To determine if the rate of change in the number of fish in the lake is increasing or decreasing at 5 A.M. (t = 5), we need to find the sign of the second derivative of N(t) at t = 5. Taking the second derivative of N(t) gives:

N''(t) = -(15π2/36)sin(πt/6) - 40.2

Plugging in t = 5, we get:

N''(5) = -(15π2/36)sin(5π/6) - 40.2

Simplifying, we get:

N''(5) ≈ -60.5

Since N''(5) is negative, the rate of change in the number of fish in the lake is decreasing at 5 A.M. (t = 5). This means that the number of fish entering the lake is decreasing faster than the number of fish leaving the lake, so the total number of fish in the lake is decreasing.

(a) Approximately 131 fish enter the lake over the 5-hour period from midnight to 5 A.M.

(b) The average number of fish that leave the lake per hour over the same period is approximately 14.8.

(c) The greatest number of fish in the lake occurs at time t = 2.94 hours, or approximately 2 hours and 56 minutes past midnight.

(d) The rate of change in the number of fish in the lake is increasing at 5 A.M.

(a) To find the total number of fish that enter the lake over 5 hours, we need to integrate the function E(t) from t=0 to t=5:

∫[0,5] E(t) dt = ∫[0,5] (20 + 15 sin(πt/6)) dt

This evaluates to approximately 131 fish.

(b) The average number of fish that leave the lake per hour can be found by calculating the total number of fish that leave the lake over 5 hours and dividing by 5:

∫[0,5] L(t) dt = ∫[0,5] (4 + 20.1t^2) dt

This evaluates to approximately 74 fish, so the average number of fish that leave the lake per hour is approximately 14.8.

(c) To find the time at which the greatest number of fish is in the lake, we need to find the maximum of the function N(t) = ∫[0,t] E(x) dx - ∫[0,t] L(x) dx over the interval [0,8]. We can do this by finding the critical points of N(t) and evaluating N(t) at those points. The critical point is at t = 2.94 hours, and N(t) is increasing on either side of this point, so the greatest number of fish is in the lake at time t = 2.94 hours.

(d) The rate of change in the number of fish in the lake at 5 A.M. can be found by calculating the derivative of N(t) at t=5. The derivative is positive, so the rate of change in the number of fish is increasing at 5 A.M.

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