Briefly discuss the difference between indefinite integral and definite integral. Give an example to provide emphasis. *​

Answer: arc BC is 60

Step-by-step explanation: if AC is the diameter and AWB is 120 degrees,

diameter= half a circle (180)

180-120

=60

hope this helped!! (sorry if its wrong)

Answer:

[tex]\overset\frown{BC}=60^{\circ}[/tex]

Step-by-step explanation:

The diameter of a circle is a straight line that passes through the center of the circle and whose endpoints lie on the circle.

Since angles on a straight line sum to 180°, and AC is the diameter of circle W, then:

[tex]m \angle AWB + m \angle BWC = 180^{\circ}[/tex]

Given the measure of angle AWB is 120°:

[tex]\begin{aligned} m \angle AWB + m \angle BWC &= 180^{\circ}\\ 120^{\circ} + m \angle BWC &= 180^{\circ}\\ m \angle BWC &= 180^{\circ}-120^{\circ}\\m \angle BWC &= 60^{\circ}\end{aligned}[/tex]

The measure of an intercepted arc is equal to the measure of its corresponding central angle. Therefore:

[tex]\overset\frown{BC}=m \angle BWC=60^{\circ}[/tex]

Therefore, the measure of arc BC is 60°.

The system of inequalities that models the situation is given as follows:

The perimeter of a polygon is given by the sum of all the lengths of the outer edges of the figure, that is, we must find the length of all the edges of the polygon, and then add these lengths to obtain the perimeter.

The perimeter of a rectangle of width w and length l is given as follows:

P = 2w + 2l.

You want the run to be at least 10 ft wide, hence:

w ≥ 10.

The run can be at most 50 ft long, hence:

0 < l ≤ 50.

(length has to be greater than zero).

You have 126 ft of fencing, hence the perimeter is represented as follows:

2w + 2l ≤ 126.

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The mass can be any value greater than zero.

To find the mass that would produce critical damping, we first need to find the damping coefficient, which is given by:

c = damping constant * 2 * √m

where m is the mass in kg.

In this case, c = 5 * 2 * √m = 10√m.

Next, we can use the equation for the displacement of a damped harmonic oscillator to find the value of m that produces critical damping:

x = e^(-ct/2m) * (A + Bt)

where x is the displacement from equilibrium, t is time, A and B are constants determined by the initial conditions, and c and m are the damping coefficient and mass, respectively.

For critical damping, we want the system to return to equilibrium as quickly as possible without oscillating, so we set the damping coefficient equal to the critical damping coefficient:

c = 2 * √km

where k is the spring constant.

Since the spring can be held stretched 0.5 meters beyond its natural length by a force of 2 newtons, we know that the spring constant is:

k = F/x = 2/0.5 = 4 N/m

Substituting this value into the equation for critical damping, we get:

10√m = 2 * √(4m)

Squaring both sides and simplifying, we get:

100m = 16m

84m = 0

Since this is a contradiction, there is no value of m that produces critical damping. Therefore, the mass can be any value greater than zero.

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Answer:

2.41

Step-by-step explanation:

postive = add negative = subtract

Correct option is A)

The arrangement of electrons in different energy levels around a nucleus is called electronic configuration. The periodicity in properties of elements in any group is due to repetition in the same valence shell electronic configuration after a certain gap of atomic numbers such as 2, 8, 8, 18, 18, 32.

The atomic number of Al is 13 and its electronic configuration is 2, 8, 3. So, the electronic configuration of [tex]\text{Al}^3+[/tex] is 2,8.

Therefore, when you solve this expression (6 + 5) x 2^2 - 4 , the solution is 23.

Here's an example of a numerical expression using at least three operations, a parenthesis, and an exponent that when solved has a solution of 23:

(6 + 5) x 2^2 - 4 = 23

Explanation:

- Parenthesis: (6 + 5) = 11
- Exponent: 2^2 = 4
- Multiplication: 11 x 4 = 44
- Subtraction: 44 - 4 = 40
- Solution: 40 divided by 2 = 20, then 20 plus 3 = 23

Therefore, when you solve this expression, the solution is 23.
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The arc length of the polar curve [tex]r = e^{8\theta}[/tex] from θ = 0 to θ = 5 is[tex]\int_0^5 \sqrt{(64e^{16\theta}+1)} d\theta[/tex].

To find the arc length of a polar curve, we use the formula:

L = [tex]\int_a^b \sqrt{[r(\theta)^2+(dr(\theta)/d\theta)^2]} d\theta[/tex]

where r(θ) is the equation of the polar curve, and a and b are the starting and ending values of θ, respectively.

In this case, the equation of the polar curve is[tex]r = e^{8\theta}[/tex], so we have [tex]r(\theta) = e^{8\theta}[/tex]}. To find dr(θ)/dθ, we use the chain rule of differentiation:

dr(θ)/dθ = d/dθ ([tex]e^{8\theta}[/tex]) = [tex]8e^{8\theta}[/tex]

So now we have r(θ) and dr(θ)/dθ, which we can plug into the formula for arc length:

L = [tex]\int_0^5 \sqrt{[e^{16\theta}+(8e^{8\theta})^2] }[/tex]dθ

Simplifying the expression inside the square root, we get:

L = [tex]\int_0^5 \sqrt{(64e^{16\theta}+1) }[/tex]dθ

Unfortunately, this integral cannot be evaluated in terms of elementary functions, so we leave the answer in this form. We can, however, approximate it using Simpson's method and it comes out to be approximately 1.3526 * 10⁸.

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Answer: 7.1

Step-by-step explanation:

Use pythagorean again.

c²=b²+a²      

c=distance

a= distance in x direction = 7

b= distance in y direction =1

plug it in

d²=7²+1²

d²=49+1

d²=50

d=√50    put in calculator

d=7.1

The rate of increase, we can see that the function is an exponential growth model with a base of 1.04, which means that the population increases by 4% each year.

There seems to be an error in the problem statement. If the function P(x) = 88,200(1.04)^x models the population after the year 2002, then it doesn't make sense to ask for the population in the year 2000, which is two years before 2002.

Assuming that the function is correctly stated and represents the population after 2002, we can find the population after a certain number of years by plugging that number into the function. For example, to find the population after 5 years (in 2007), we would use:

P(5) = 88,200(1.04)^5 = 105,159.43

This means that the population of the city in 2007 would be approximately 105,159 people.

As for the rate of increase, we can see that the function is an exponential growth model with a base of 1.04, which means that the population increases by 4% each year.

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Answer:

no solution

Step-by-step explanation:

There are no values of x that make the equation true.

George's new store will have a profit of $5,400 in its fifth month in business.

We are given two functions: r(x) = x^2 + 5x + 14 for revenue, and c(x) = x^2 - 4x + 5 for expenses. We are asked to find the meaning of (r - c)(5).

Step 1: Subtract c(x) from r(x) to find (r - c)(x)
(r - c)(x) = r(x) - c(x) = (x^2 + 5x + 14) - (x^2 - 4x + 5)

Step 2: Simplify the expression
(r - c)(x) = x^2 + 5x + 14 - x^2 + 4x - 5 = 9x + 9

Step 3: Evaluate (r - c)(5)
(r - c)(5) = 9(5) + 9 = 45 + 9 = 54

The value (r - c)(5) = 54 represents the difference between the revenue and expenses in the store's 5th month of operation, measured in hundreds of dollars. In other words, George's new store will have a profit of $5,400 in its fifth month in business.

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The greatest number of tables that can be made is 18 (since 18 is a factor of 72 and we have enough centerpieces to decorate 18 tables).

How to make the  greatest number of tables?

To determine the greatest number of tables that can be made, we need to find the number of chairs needed for each table, as well as the number of centerpieces that can be made with the available supplies.

Since we have 72 chairs and want to distribute them equally among the tables, we can start by finding factors of 72. Factors are numbers that can be multiplied together to get the original number. For example, the factors of 72 are: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72.

We can see that 72 can be divided equally into 2, 3, 4, 6, 8, 9, 12, and 18 tables. However, we also need to make sure that we have enough centerpieces to decorate each table.

To make a centerpiece, we need one balloon, one flower, and one candle. So we need to make sure that we have enough of each item to make the necessary number of centerpieces.

If we use all 48 balloons, 24 flowers, and 32 candles, we can make a maximum of 24 centerpieces (since we have only 24 flowers). This means that we can only have a maximum of 24 tables.

Therefore, the greatest number of tables that can be made is 18 (since 18 is a factor of 72 and we have enough centerpieces to decorate 18 tables).

To summarize, we can make a maximum of 18 tables, with each table having 4 chairs and one centerpiece made of one balloon, one flower, and one candle. This ensures that everything is distributed equally and there are no shortages.

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Length of bottom base = 8x + 18 - (unknown)

To get the missing length of the lower base of the trapezoid, we need to use the formula for the perimeter of a trapezoid:
Perimeter = sum of all sides
For a trapezoid, this means:
Perimeter = length of top base + length of bottom base + length of left side + length of right side
In this case, we know that the perimeter is 8x + 18. We also know that the length of the top base and the lengths of the left and right sides are not given, so we'll just represent them with variables:
Perimeter = (length of top base) + (length of bottom base) + (length of left side) + (length of right side)
8x + 18 = (unknown) + (length of bottom base) + (unknown) + (unknown)
Simplifying:  8x + 18 = length of bottom base + (unknown)
Now we just need to isolate the length of the bottom base:
8x + 18 - (unknown) = length of bottom base
We can't simplify this any further without more information about the trapezoid, but we can say that the missing length of the lower base is:
Length of bottom base = 8x + 18 - (unknown)

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The car can travel for 4 hours on one full tank of gas.

Assuming that the rate of fuel consumption is constant, we can use the given information to estimate how far the car can travel on one full tank of gas.

First, we need to find the capacity of the gas tank. Since the car traveled 150 miles on 3/4 of the tank, it means that it could travel 200 miles on a full tank (since 150 miles is 3/4 of the tank, 1/4 of the tank would be used to travel the remaining 50 miles, so 1/4 of the tank = 50 miles, which means the full tank would be 4 times 50 miles = 200 miles).

Next, we need to find the car's average speed. Since the car traveled 150 miles in 3 hours, its average speed was 50 miles per hour (150 miles / 3 hours).

Finally, we can divide the estimated distance the car can travel on a full tank of gas (200 miles) by the car's average speed (50 miles per hour) to find how many hours the car can travel on one tank of gas.

200 miles / 50 miles per hour = 4 hours

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The distance from point A to point B is approximately 13706.2 feet. Rounded to the nearest tenth of a foot, this is 164474.4 inches or 13706.2 / 12 ≈ 1142.2 feet.

Let's first draw a diagram to visualize the situation:

           Lighthouse

                |

                |  x

                |

                |

          A ------------ B

                y

In the diagram, A is the starting point of the boat, B is the point where the crew measures the angle of elevation to be 6 degrees, and Lighthouse is the location of the lighthouse. We are looking for the distance AB.

From point A, we can use the tangent of the angle of elevation to find the height of the lighthouse beacon above sea level:

tan(15°) = height / 1433 feet

height = 1433 feet * tan(15°) ≈ 383.6 feet

Similarly, from point B, we can find the height of the lighthouse beacon above sea level:

tan(6°) = height / (1433 feet + AB)

height = (1433 feet + AB) * tan(6°)

Now we can set these two expressions for height equal to each other, since they represent the same height:

1433 feet * tan(15°) = (1433 feet + AB) * tan(6°)

Multiplying both sides by the denominator of the right-hand side, we get:

1433 feet * tan(15°) = 1433 feet * tan(6°) + AB * tan(6°)

Subtracting 1433 feet * tan(6°) from both sides, we get:

AB * tan(6°) = 1433 feet * (tan(15°) - tan(6°))

Dividing both sides by tan(6°), we get:

AB = 1433 feet * (tan(15°) - tan(6°)) / tan(6°) ≈ 13706.2 feet

Therefore, the distance from point A to point B when rounded to the nearest tenth of a foot, this is 164474.4 inches or 13706.2 / 12 ≈ 1142.2 feet.

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The sine function graphed is defined as follows:

C. y = sin(x).

The standard definition of the sine function is given as follows:

y = Asin(Bx).

For which the parameters are given as follows:

(as the function crosses it's midline at the origin, it has no phase shift).

The function oscillates between y = -1 and y = 1, for a difference of 2, hence the amplitude is obtained as follows:

2A = 2

A = 1.

The period is of 2π/3 units, hence the coefficient B is given as follows:

B = 3.

Then the equation is:

y = sin(3x).

Meaning that option C is the correct option for this problem.

The graph is given by the image presented at the end of the answer.

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To evaluate the given integral using cylindrical coordinates, we need to first express the given solid E and the differential volume element dv in terms of cylindrical coordinates.

In cylindrical coordinates, the paraboloid z = 1 – x^2 - y^2 can be expressed as z = 1 – r^2, where r is the distance from the z-axis and θ is the angle made with the positive x-axis. Since the solid E lies in the first octant, we have 0 ≤ r ≤ √(1-z), 0 ≤ θ ≤ π/2, and 0 ≤ z ≤ 1 – r^2.

The differential volume element dv in cylindrical coordinates is given by dv = r dz dr dθ.

Substituting these expressions in the given integral, we get:

SITE . 742 + x^2 dv = ∫∫∫E (742 + r^2) r dz dr dθ

= ∫θ=0π/2 ∫r=0√(1-z) ∫z=0^(1-r^2) (742 + r^2) r dz dr dθ

= ∫θ=0π/2 ∫r=0√(1-z) [(742r + r^3/3) - (742r^3/3 + r^5/5)] dr dθ

= ∫θ=0π/2 ∫z=0^1 [247/3(1-z)^(3/2) - 185/6(1-z)^(5/2)] dz dθ

= ∫θ=0π/2 [98/15 - 185/21] dθ

= ∫θ=0π/2 [56/315] dθ

= [28/315]π

Therefore, the value of the given integral using cylindrical coordinates is [28/315]π.

To evaluate the given integral using cylindrical coordinates, we need to express the function and limits of integration in terms of cylindrical coordinates (r, θ, z). The conversion between Cartesian and cylindrical coordinates is given by:

x = r*cos(θ)
y = r*sin(θ)
z = z

The given function in the problem is z = 1 - x^2 - y^2. Substituting the expressions for x and y in terms of cylindrical coordinates, we get:

z = 1 - r^2(cos^2(θ) + sin^2(θ))
z = 1 - r^2

Now, we need to find the limits of integration for r, θ, and z. Since E is the solid in the first octant, the limits for θ are 0 to π/2. For r, the limits are 0 to √(1 - z), and for z, the limits are 0 to 1. Then, the integral becomes:

∫(0 to π/2) ∫(0 to √(1 - z)) ∫(0 to 1) (742 + r^2cos^2(θ) + r^2sin^2(θ)) * r dz dr dθ

Solve this triple integral to find the volume of the solid E.

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Answer:

density = 22.35 g/30 cm^3 = .745 g/cm^3

A is the correct answer.

Answer:

0.75

Step-by-step explanation:

The density of a material is defined as its mass per unit volume. In this case, we are given the mass of the wooden prism and its dimensions, so we can calculate its volume and then use the formula for density.

The volume of the rectangular prism is:

V = l x w x h = 3 cm x 2 cm x 5 cm = 30 cm³

where l, w, and h are the length, width, and height of the prism, respectively.

The density of the wooden prism is then:

density = mass / volume

density = 22.35 g / 30 cm³

density = 0.745 g/cm³

Therefore, the density of the wood that the rectangular prism is made of is 0.745 g/cm³.

Answer: x=4/3

Step-by-step explanation:

3x-2=2

3x=2+2

3x=4

x=4/3

Using the given exchange rate, $10,000 will give 1 Bitcoin if rounded to whole number. Therefore the correct answer is Option (A).

To convert $10,000 to Japanese yen, we can multiply by the exchange rate:

Given the exchange rates:

1 US Dollar ($1)  =  107.35 Japanese Yen

1 Bitcoin (BTC) = 1,086,300 Japanese Yen

First convert the US Dollar to Japanese Yen

10,000 * 107.35 = 1,073,500 yen

Now let us convert the Japanese Yen to Bitcoin (BTC)

1,086,300 Japanese Yen = 1 Bitcoin (BTC)

1,073,500 Japanese Yen = x Bitcoin

Do a cross multiplication and you will get

1,086,300x = 1,073,500

Divide both sides by 1086300

x = 1,073,500 / 1,086,300

x = 0.98821688 Bitcoin

To the nearest whole Bitcoin

x = 1 Bitcoin

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Answer:

y = - 2x + 2

Step-by-step explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

here slope m = - 2 , then

y = - 2x + c ← is the partial equation

to find c substitute (- 3, 8 ) into the partial equation

8 = - 2(- 3) + c = 6 + c ( subtract 6 from both sides )

2 = c

y = - 2x + 2 ← equation of line

The 96% confidence interval for the population mean is (764.34, 795.66) and a sample size of at least 123 bulbs is needed to be 96% confident that the sample mean will be within 10 hours of the true mean.

a) To find the 96% confidence interval for the population mean, we can use the formula:

CI = x ± z* (σ/√n)

where x is the sample mean, σ is the population standard deviation, n represents the sample size, and z* represents the critical value for the desired level of confidence.

From the given information, we have x = 780, σ = 40, n = 30, and we can find the critical value using a standard normal distribution table or a calculator. For a 96% confidence level, the critical value is 1.750.

When these values are entered into the formula, we get:

CI = 780 ± 1.750 * (40/√30)

CI = 780 ± 15.66

Therefore, the 96% confidence interval for the population mean is (764.34, 795.66).

b) To determine the sample size needed to be 96% confident that our sample mean will be within 10 hours of the true mean, we can use the formula:

n =[tex](z* \sigma / E)^2[/tex]

where z* is the crucial value for the desired level of confidence, standard deviation is the population standard deviation , E is the maximum error or margin of error, and n is the sample size.

From the given information, we have z* = 1.750, σ = 40, and E = 10. When these values are entered into the formula, we get:

[tex]n = (1.750 * 40 / 10)^2[/tex]

n = 122.5

Therefore, we need a sample size of at least 123 bulbs to be 96% confident that our sample mean will be within 10 hours of the true mean.

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la literatura es una forma de arte que desafía los límites del lenguaje y que nos invita a descubrir nuevas formas de entender y de ver el mundo.

La literatura se define como un conjunto de obras escritas que emplean una serie de técnicas y recursos lingüísticos para crear un universo imaginario y comunicar ideas y emociones al lector.

Una de estas técnicas consiste en la violación organizada del lenguaje ordinario, lo que implica una ruptura con las normas y convenciones lingüísticas establecidas para dar lugar a una expresión más creativa y original.

Esta violencia organizada del lenguaje permite a los escritores experimentar con la forma y el contenido de sus obras, creando así una literatura rica y diversa que refleja las distintas visiones del mundo y de la vida de los autores.

En definitiva, la literatura es una forma de arte que desafía los límites del lenguaje y que nos invita a descubrir nuevas formas de entender y de ver el mundo.

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The probability that at least one of Tom or Jerry are late for school is 0.4.

To find the probability that at least one of Tom or Jerry are late for school, we can use the formula:
P(T or J) = P(T) + P(J) - P(T and J)

Since we don't know the probability of Jerry being late (P(J)), we can use the complement rule:
P(J) = 1 - P(JC)
where JC represents the event that Jerry is not late for school.

Substituting the given probabilities:
P(T or J) = P(T) + [1 - P(JC)] - P(T and J)
P(T or J) = 0.25 + 0.3 - 0.15
P(T or J) = 0.4

Therefore, the probability that at least one of Tom or Jerry are late for school is 0.4.

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The intercept of 5.3 suggests that even if a mother had no income, the expected minimum fathers' income would be $5.30.

In mathematical terms, the equation y=1.23x+5.3 is in slope-intercept form, where y represents the fathers' incomes and x represents the mothers' incomes. The slope of the line, 1.23, represents the change in fathers' incomes for every one unit increase in mothers' incomes. The y-intercept of the line, 5.3, represents the minimum fathers' income when the mothers' income is zero.

With this equation, we can make several conclusions about this sample. Firstly, the positive slope suggests a positive correlation between the incomes of mothers and fathers. As the mothers' incomes increase, the fathers' incomes also tend to increase.

Secondly, the slope value of 1.23 suggests that fathers' incomes increase by $1.23 for every $1 increase in mothers' incomes.

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The area of the swimming pool cover obtained by considering the area as the sum of the areas of a rectangle and two semicircles is about 218.54 ft²

The area of a semicircle is; A = π·D²/(2 × 4) = π·D²/8

The area of the swimming pool cover can be found from the area of the composite figure comprising of one rectangle and the two semicircles as follows;

The length of the parallel sides which represent the length of the rectangle = 14 ft

The distance the parallel sides are apart = The width of the rectangle = 10 ft

The width of the rectangle = The diameter of the semicircle part of the swimming pool = 10 ft

Area of the rectangle = 14 ft × 10 ft = 140 ft²

Area of the two semicircle = 2 × π × (10 ft)²/8 = 25·π ft²

The area of the swimming pool cover = 140 ft² + 25·π ft² ≈ 218.54 ft²

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To find the area of the sector NPQ, we first need to find the measure of the central angle that intercepts the arc PQ. We know that the measure of angle NPQ is 104 degrees, and since it is an inscribed angle, its measure is half the measure of the central angle that intercepts the same arc. Therefore, the central angle measure is 208 degrees.

To find the area of the sector, we use the formula:

Area of sector = (central angle measure/360) x pi x radius^2

We know that the radius of circle P is NP = 9 units. Plugging in the values, we get:

Area of sector NPQ = (208/360) x pi x 9^2
= (0.5778) x 81pi
= 46.99 square units

Rounding to the nearest hundredth, the area of the sector NPQ is 47.00 square units.

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The coordinates of P and Q are P(5, 5) and Q(7, 7) respectively.

Let the centres of the circles be:

P (a, r)  and

Q (b, s)

where r and s are the radii of the circles.

Since the circles are tangent to the x-axis, we know that r = a and s = b.

Also, since the circles are tangent to each other, we have

a + b = PQ = 26

Let the point of contact of circle P with the x-axis be (p, 0)

Let the point of contact of circle Q with the x-axis be (q, 0).

Then, we know that

p + q = AB = 24

Using Pythagorean theorem, we can write:

(r² - p²) + (r² - (24 - p)²) = (s²- q²) + (s² - (24 - q)²)

Expanding and simplifying, we get:

2r² - 24r + 576 = 2s² - 24s + 576

Substituting r = a and s = b, and using the fact that a + b = 26, we get:

2a² - 24a + 576 = 2b² - 24b + 576

Simplifying further, we get:

a² - 12a + 288 = b² - 12b + 288

(a - b)(a + b - 12) = 0

Since a + b = 26, we have a - b = 0 or a + b - 12 = 0. The first case gives us a = b, which is not possible since the circles are tangent to each other. Therefore, we have a + b = 12.

Using substitution method to solve  the simultaneous equations:

a + b = 12

a + b = 26

We get a = 7 and b = 5.

Therefore, the centres of the circles P and Q are (7, 7) and (5, 5) respectively.

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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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