-11 +23-46+ (-12)
how to solve it​

Given the demand function P=$6.00-0.100(Qd)

a) at a price of $2.00, the quantity demanded is 40 units, thus, Total Revenue = $80

b) at a price of $1.00, the quantity demanded is 50 units, thus, Total Revenue  = $50

c) Since the price elasticity of demand is less than 1, demand is price inelastic between these two prices.

Given the demand function P=$4.80-0.060(Qd)

d)
The optimal price to maximize total revenue is $2.40 and the optimal quantity is 40 units. TR = $96

e) When the price is $3.00 and the quantity demanded is 30, TR = $90

f) When the price is $1.80 and the quantity demanded is 50, the total revenue = $90

g) If demand is price elastic and the market price is $3.00, total revenue can be increased by reducing the price.

A demand function is a mathematical equation that describes the relationship between the price of a good or service and the quantity of that good or service that consumers are willing and able to purchase at that price.

a. When the price is $2.00, the quantity demanded can be calculated by setting Qd equal to 2.00 in the demand equation:

Qd = (6.00 - 0.100Qd)

2.00 = (6.00 - 0.100Qd)

0.100Qd = 4.00

Qd = 40

Therefore, at a price of $2.00, the quantity demanded is 40 units. The total revenue can be calculated by multiplying the price by the quantity demanded:

Total Revenue = Price x Quantity Demanded = 2.00 x 40 = $80

b. When the price is $1.00, the quantity demanded can be calculated in the same way:

Qd = (6.00 - 0.100Qd)

1.00 = (6.00 - 0.100Qd)

0.100Qd = 5.00

Qd = 50

Therefore, at a price of $1.00, the quantity demanded is 50 units. The total revenue can be calculated as:

Total Revenue = Price x Quantity Demanded = 1.00 x 50 = $50

c. To determine the price elasticity of demand between these two prices, we need to compare the percentage change in quantity demanded to the percentage change in price.

At a price of $2.00, the quantity demanded is 40 units. At a price of $1.00, the quantity demanded is 50 units. The percentage change in quantity demanded can be calculated as:

% Change in Quantity Demanded = (New Quantity Demanded - Old Quantity Demanded) / Old Quantity Demanded x 100%

= (50 - 40) / 40 x 100% = 25%

The percentage change in price can be calculated as:

% Change in Price = (New Price - Old Price) / Old Price x 100%

= (1.00 - 2.00) / 2.00 x 100% = -50%

The price elasticity of demand between these two prices can be calculated as:

Price Elasticity of Demand = % Change in Quantity Demanded / % Change in Price

= 25% / -50%

= -0.5

Since the price elasticity of demand is less than 1, demand is price inelastic between these two prices. This means that a change in price will result in a proportionately smaller change in quantity demanded.

d). To find the optimal price and quantity to maximize total revenue, we need to take the derivative of the total revenue function with respect to quantity and set it equal to zero.

Total Revenue = Price x Quantity Demanded

TR = (4.80 - 0.060Qd)Qd

TR = 4.80Qd - 0.060Qd^2

Taking the derivative of TR with respect to Qd:

dTR/dQd = 4.80 - 0.120Qd

Setting dTR/dQd equal to zero:

4.80 - 0.120Qd = 0

Qd = 40

Substituting Qd = 40 into the demand equation to find the optimal price:

P = 4.80 - 0.060(40)

P = 2.40

Therefore, the optimal price to maximize total revenue is $2.40 and the optimal quantity is 40 units. The total revenue can be calculated as:

TR = P x Q = 2.40 x 40 = $96

e. When the price is $3.00 and the quantity demanded is 30, the total revenue can be calculated as:

TR = P x Q = 3.00 x 30 = $90

f. When the price is $1.80 and the quantity demanded is 50, the total revenue can be calculated as:

TR = P x Q = 1.80 x 50 = $90

g. If demand is price elastic and the market price is $3.00, total revenue can be increased by reducing the price. This is because the percentage increase in quantity demanded will be greater than the percentage decrease in price, resulting in a net increase in total revenue.


Therefore, the company could consider lowering the price from $3.00 to a price closer to the optimal price of $2.40 to increase total revenue.

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The answer of the given question based on the circle on finding the measure of the manilla angle the answer will be  the measure of the manila angle is 110° degrees.

A circle is  closed, two-dimensional shape that consists of all points in  plane that are equidistant from  given point, called center of the circle. The distance from  center to any point on circle is called  radius of the circle.

The circumference of  circle is distance around the circle, and it is equal to 2π times radius. Circles are fundamental concept in geometry and are used in many fields, like physics, engineering, and architecture. They are also used to represent cycles, cycles of time, and the cyclical nature of life.

If the entire pink arc measures 200° degrees and the entire blue arc measures 50° degrees, then the sum of the measures of the three arcs (pink, blue, and manila) is 360° degrees, which is the total measure of a circle.

To find the measure of the manila angle, we need to subtract the measures of the pink and blue arcs from 360° degrees:

360° degrees - 200° degrees - 50° degrees = 110° degrees

Therefore, the measure of the manila angle is 110° degrees.

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Answer:

115

Step-by-step explanation:

5 times 23= answer

Answer:

Were sorry! Answer is not available right now check in later.

Step-by-step explanation:

a) To find the number of different first names President Snow could create using the letters in "Coriolanus", we can use the formula for permutations with repeated letters. There are 10 letters in "Coriolanus", but "o" and "i" each appear twice, so the total number of permutations is:

10! / (2! * 2!) = 45,360 / 4 = 11,340

Therefore, President Snow could create 11,340 different first names using the letters in "Coriolanus".

b) If the consonants in "Coriolanus" are kept together, we have "Crlns" as a string of consonants. This gives us 5 consonants to arrange, so the number of permutations is:

5! = 120

Therefore, President Snow could create 120 different first names using the consonants in "Coriolanus" kept together.

c) To count the number of different first names using "Coriolanus" that end with a vowel, we can consider the last letter of the name. There are 4 vowels in "Coriolanus", so there are 4 choices for the last letter. For the other letters, we can use the remaining 9 letters (excluding the last vowel) in any order. Therefore, the number of different first names that end with a vowel is:

4 * 9! = 1451520

Therefore, President Snow could create 1,451,520 different first names using "Coriolanus" that end with a vowel.

d) If the consonants in "Coriolanus" are kept in alphabetical order, then we have "aclnorsu". This gives us 8 letters to arrange, so the number of permutations is:

8! = 40,320

Therefore, President Snow could create 40,320 different first names using the consonants in "Coriolanus" in alphabetical order.

e) To find the number of different first names President Snow could create without repeating any letters, we can use the formula for permutations without repetition. There are 10 letters in "Coriolanus", so the total number of permutations is:

10! = 3,628,800

Therefore, President Snow could create 3,628,800 different first names without repeating any letters.

a) To find the number of ways to select 2 youths from the group of eligible youths, we can use the formula for combinations. We have 64 boys and 59 girls, so the total number of eligible youths is 64 + 59 = 123. The number of ways to select 2 youths is:

123C2 = (123 * 122) / 2 = 7503

Therefore, there are 7,503 ways to select 2 youths from the entire group of eligible youths.

b) To find the number of ways to select a single boy and girl as "tributes" from the eligible youths, we can use the product rule. There are 64 boys to choose from and 59 girls to choose from, so the number of ways to select one boy and one girl is:

64 * 59 = 3,776

Therefore, there are 3,776 ways to select a single boy and girl as "tributes" from the eligible youths.

c) To find the number of ways to select a girl first followed by her brother, we can use the product rule again. There are 59 girls to choose from for the first selection, and after a girl is selected, there are 63 youths left to choose from (excluding the selected girl and the 14 pairs of brother and sister groups). Therefore, the number of ways to select a girl first followed by her brother is:

59 *

Answer:

29

Step-by-step explanation:

First, let's find out how many members are added to the squad with the new coach:

45% of 20 = 0.45 x 20 = 9

So, with the new coach, there are 9 more members added to the squad.

Now, to find the total number of members on the squad with the new coach, we just need to add the old number of members (20) to the number of new members added (9):

20 + 9 = 29

Therefore, there are now 29 members on the cheerleading squad at Morristown High School with the new coach.

Answer: false

Step-by-step explanation:

Answer: 180,000 g

Step-by-step explanation:

1st add all masses together:

80kg + 40kg + 60 kg = 180 kg

then we need to convert kilograms to grams:

1 kg = 1000g

180kg * (1000g / 1kg) = 180,000 g

To calculate the total mass (in grams) of the three boxes, we first need to convert their masses from kilograms to grams and then add them together.

The mass of the first box is 80 kg, which is equivalent to 80,000 grams (because 1 kg = 1000 grams).

The mass of the second box is 40 kilograms, or 40,000 grams.

The mass of the third box is 60 kilograms, or 60,000 grams.

To find the total mass of the three boxes, we add these values:

80,000 grams + 40,000 grams + 60,000 grams = 180,000 grams

Therefore, The total mass of the three boxes in gram is 180,000

The correct answer is D) 180,000

Showing solidarity and discipline in protecting our environment requires collective effort and a willingness to change our behavior and attitudes towards the environment. Here are some ways in which we can show solidarity and discipline in protecting our environment:

Educate ourselves and others about environmental issues and their impact on our planet and communities.

Reduce, reuse, and recycle waste by properly disposing of garbage, avoiding single-use plastics, and using eco-friendly products.

Conserve resources like water and energy by taking shorter showers, turning off lights and electronics when not in use, and using public transportation or carpooling.

Plant trees, flowers, and other vegetation to improve air quality, provide shade, and prevent erosion.

Support environmental causes and organizations through volunteering, donating, and spreading awareness.

By taking these actions, we can demonstrate our solidarity and discipline in protecting our environment, and inspire others to join us in making a positive impact on the planet.

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The hotel served 306 gallons of orange juice this year.

To find the amount of orange juice served this year, we need to add 70% more of the amount served last year to the amount served last year. Let's denote the amount served last year as "x". Then we can set up the equation:

Simplifying this equation gives us:

We know from the problem that the amount served last year was 180 gallons. Plugging this into our equation, we get:

Simplifying this equation gives us:

Therefore, the hotel served 306 gallons of orange juice this year.

In summary, we used the information given in the problem to set up an equation and solve for the amount of orange juice served this year. We first found the amount served last year, and then added 70% more of that amount to get the total amount served this year.

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Answer: x = 6

Step-by-step explanation:

(7x+3)+85+50 = 180

(7x+3)+135 = 180

7x+3 = 180 - 135 = 45

7x = 45-3 = 42

x = 42 / 7 = 6

x = 6

The volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane x + 2y + 3z = 3 is 12 cubic units.

To calculate the volume, we can use the formula V = l × w × h, where l = length, w = width and h = height.

The three faces lie on the coordinate planes, so we can use the equation x + 2y + 3z = 3 to find the coordinates of the vertex. We can solve for the values of x, y, and z and plug them into the formula.

We will use the substitution method:

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The following is the recursive formula for the arithmetic sequence in this issue:

c(1) = -16.

c(n) = c(n - 1) - 17.

An arithmetic sequence is a series of numbers where each term is obtained by adding a fixed constant, known as the common difference, to the previous term. For example, in the sequence 2, 5, 8, 11, 14, 17, each term is obtained by adding 3 to the previous term.

The formula for finding the nth term of an arithmetic sequence is: a(n) = a(1) + (n-1)d, where a(1) is the first term, d is the common difference, and n is the term number. For example, to find the 10th term of the sequence 2, 5, 8, 11, 14, 17, we would use the formula a(10) = 2 + (10-1)3 = 29. Arithmetic sequences have many practical applications, such as in finance, where they can be used to calculate the interest earned on an investment over time.

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The solutions for the systems of inequalities are:

a) (0, -50), (0, -100), (0, -125)

b)  (0, 20), (0, 23) , (0, 24).

When we have a system of inequalities, a solution is a point (x, y) that solves both ienqualities at the same time.

The first one is:

y ≤ x - 8

y < -3x - 9

Here y must be smaller than x, then we can define x like x = 0, and really small values for y, like y = -50, replacing that we will get:

-50  ≤ 0 - 8 = -8

-50 < - 3*0 - 9 = -9

Both of these are true, so (0, -50) is a solution, and trivially, other solutions of the system of inequalities can be things like (0, -100) and (0, -125) are other two solutions.

For the second system:

y > 5x + 1

y > 3

Let's do the same thing, x = 0 and y gets really large values, like y = 20

20 > 5* + 1 = 1  this is true.

20 > 3   this is true.

so (0, 20) is a solution, and also are (0, 23) and (0, 24).

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Answer:

x = 24/19

Step-by-step explanation:

Let's use algebra to solve this problem.

Let's say the number we're looking for is x.

According to the problem, "Some number minus 24" can be written as "x - 24".

And "the product of the number and -18" can be written as "x*(-18)" or simply "-18x".

So the equation we need to solve is:

x - 24 = -18x

We can solve for x by adding 18x to both sides of the equation:

x + 18x - 24 = -18x + 18x

Simplifying the left side of the equation gives:

19x - 24 = 0

Then we can add 24 to both sides of the equation to isolate the x term:

19x - 24 + 24 = 0 + 24

Simplifying the left side of the equation gives:

19x = 24

Finally, we can solve for x by dividing both sides of the equation by 19:

x = 24/19

The solution to the system of equations is (x, y) = (1, -1) where the given equations are 11x+9y=-20 and x=-5y-6.

The substitution method is a technique used in algebra to solve systems of equations by replacing one variable with an expression containing another variable. The goal is to eliminate one of the variables so that we can solve for the other one.

We are given the following system of two equations with two variables:

11x + 9y = -20 (equation 1)

x = -5y - 6 (equation 2)

To solve the system using the substitution method, we need to solve one of the equations for one of the variables, and then substitute the expression for that variable into the other equation. Let's solve equation 2 for x:

x = -5y - 6

Now we can substitute this expression for x into equation 1, and solve for y:

11x + 9y = -20

11(-5y - 6) + 9y = -20 (substituting x = -5y - 6)

-55y - 66 + 9y = -20

-46y = 46

y = -1

Now that we have found y = -1, we can substitute this value back into equation 2 and solve for x:

x = -5y - 6

x = -5(-1) - 6

x = 1

Therefore, the solution to the system of equations is (x, y) = (1, -1).

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The number of students on the bus is 40.

The first step is to form two equations that represent the information in the question:

a + b = 45 equation 1

375a + 200b = 9875 equation 2

Where:

a = number of adults in the bus

b = number of students in the bus

The elimination method would be used to determine the number of students on the bus.

Multiply equation 1 by 375

375a + 375b = 16,875 equation 3

Subtract equation 2 from equation 3

175b = 7000

Divide both sides of the equation by 175

b = 7000 / 175

b = 40

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Answer:

|2x + 1| = 7

|2x − 1| = 7

2x + 1 = -7 or 2x + 1 = 7

Step-by-step explanation:

or if u have no time

C, E, F.

These might not be in order but hey they are correct amiright :/

Minimum radius of convergence R of power series solutions about the ordinary point x = 1 is 1.

The minimum radius of convergence R of power series solutions about the ordinary point x = 1 of the differential equation (x²-9)y''+3xy'+y=0 can be determined as follows:Let us first write the differential equation as: y''+ (3x/(x²-9)) y' + (1/(x²-9)) y= 0Therefore, we have the following properties: p(x) = (3x/(x²-9)), q(x) = (1/(x²-9)), and x0 = 1. Since p(x) and q(x) are both rational functions, and are defined for all x except ±3, the point x0 = 1 is an ordinary point.

To obtain power series solutions about the ordinary point x = 1, we should look for solutions of the form: y = (x - 1)²Σn≥0 an(x - 1)^n, where an's are constants to be determined. This is the power series solution about x0, expanded around x0, which is x = 1.Let us now plug in this series for y, y', and y'' into the differential equation:Σn≥0 an(n+1) (x - 1)^n + 3Σn≥0 an(x - 1)^(n+1) / (x - 1)² - Σn≥0 an(x - 1)^n / (x - 1)²= 0Multiplying everything by (x - 1)² and grouping together all coefficients of the same power, we have:Σn≥0 [(n+1)(n-2) an - 3an] (x - 1)^n= 0

Comparing the coefficients of the like powers of (x - 1), we get:2a1 = 0, and [(n+1)(n-2) an - 3an] = 0 for n ≥ 2.The solution for a1 = 0 does not affect the radius of convergence. Therefore, for n ≥ 2, we obtain an = 3 / [n(n-3)] by solving the quadratic equation, n² - n - 6 = 0.Therefore, the power series solution about x0 = 1 is: y = Σn≥2 [3 / (n(n-3))] (x - 1)² (x - 1)^nThe radius of convergence of this series is given by:R = limn→∞ |an / an+1|= limn→∞ |n(n-3) / (n+1)(n-2)|= 1Therefore, the minimum radius of convergence of the power series solutions about the ordinary point x = 1 is R = 1.

Hence, the answer is: Minimum radius of convergence R of power series solutions about the ordinary point x = 1 is 1.

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Two teaspoons of salt contains 7.16 grams of Na or 7160 milligrams of Na.

Given that a soup recipe calls for two teaspoons of salt. We need to find out how many milligrams of sodium is that?

1 teaspoon = 5.69 grams      1 gram = 1000 milligrams

2 teaspoons of salt = 2 * 5.69 grams = 11.38 grams of salt

11.38 grams of salt = 11.38 * 1000 milligrams = 11380 milligrams of salt

Now, we have to find out how much sodium (Na) is there in 11380 milligrams of salt. Sodium chloride is the chemical name for table salt (NaCl). So, the atomic mass of NaCl can be calculated as follows:

Na = 1Cl = 35.45

Atomic mass of NaCl = Na + Cl= 1 + 35.45= 36.45

So, 1 mole of NaCl = 36.45 grams         11380 milligrams of NaCl = 11380/1000 grams= 11.38/36.45 moles

Therefore, Moles of Na = 11.38/36.45 = 0.3121

mol Atomic mass of Na is 22.99 g/mol.

So, 1 mole of Na weighs = 0.3121 * 22.99= 7.16 grams

Therefore, 11380 milligrams of NaCl = 7.16 grams of Na. Hence, two teaspoons of salt contains 7.16 grams of Na or 7160 milligrams of Na.

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The car must travel at an average speed of 105 km/h to cover the same distance in 3 hours and 30 minutes.

Let's start by calculating the distance covered by the car in 3 hours 40 minutes.

3 hours 40 minutes is equal to 3.67 hours.

Distance = Speed × Time

Distance = 100 km/h × 3.67 h

Distance = 367 km

Now, let's find the average speed the car must travel to cover the same distance in 3 hours and 30 minutes.

3 hours and 30 minutes is equal to 3.5 hours.

Distance = Speed × Time

367 km = Speed × 3.5 h

Speed = Distance / Time

Speed = 367 km / 3.5 h

Speed = 105 km/h

Therefore, the car must travel at an average speed of 105 km/h to cover the same distance in 3 hours and 30 minutes.

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 In conclusion, the Poisson distribution was successfully applied by Student (1907) to the study of errors in counting yeast cells or blood corpuscles with a haemacytometer. It is possible to calculate an approximate 95% confidence interval for each estimated count, as well as to compare observed and expected counts.

The Poisson distribution was first applied to the study of errors made in counting yeast cells or blood corpuscles with a haemacytometer by Student (1907). The study involved the preparation of four different concentrations of a mixture of yeast cells, water, and gelatin spread on a glass. Counts were made on 400 squares and the data summarized in the following table.

An approximate 95% confidence interval for each estimate can be calculated using the Poisson distribution. For each of the four concentrations, the lower bound of the confidence interval is given by the formula x - 1.96*sqrt(x) and the upper bound is given by the formula x + 1.96*sqrt(x), where x is the observed count for that concentration.

It is also possible to compare the observed counts with the expected counts for each concentration. The expected count for each concentration is given by the formula λ = n*p, where n is the number of squares and p is the probability of an event occurring in a single square. The expected counts can be compared to the observed counts to determine whether they are in agreement with the Poisson distribution.

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Answer:

5.333 hours

Step-by-step explanation:

We know

4 Pipes fill a tank in 16 hours.

How long will it take to fill the tank if 12 pipes of the same dimensions are used?

We Take

16 x 1/3 = 5.333 hours

So, it takes about 5.333 hours to fill the tank.

Ryan makes a profit of £240.  the total sales now amount to £600. By subtracting the cost of the jumpers, i.e. £190, from the total sales, we calculate that Ryan makes a profit of £240.

Ryan spends £190 to buy 80 jumpers. He sells 50% of the jumpers, i.e. 40 jumpers, at £12 each. This brings the total sales to £480. Then, he puts the remaining 40 jumpers on a Buy one get one half price offer. He sells 20 of the remaining jumpers using this offer. Therefore, the total sales now amount to £600. By subtracting the cost of the jumpers, i.e. £190, from the total sales, we calculate that Ryan makes a profit of £240.

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The standard error of the mean is defined as the standard deviation of the sampling distribution of all potential sample means drawn from a population.

The standard error is also known as the standard deviation of the sampling distribution.

The standard deviation of the population is known as the population standard deviation.

The population standard deviation is an estimate of the amount of variation or dispersion of the values in the population.

It represents the average distance of the values from the mean of the population.

The standard deviation of the sampling distribution of all possible sample means (for a fixed sample size n) from a population is lower than the standard deviation of the population.

The standard deviation of the sampling distribution of all possible sample means is a measure of the variation of the sample means around the population mean.

The sample means are less variable than the individual observations in the population; therefore, the standard deviation of the sample means is lower than the standard deviation of the population.

In general, the standard error is given by:

$$SE = \frac{{{\sigma _p}}}{{\sqrt n }}$$

where, σp represents the population standard deviation and n represents the sample size.

The standard deviation of the sampling distribution is lower than the population standard deviation.

The standard deviation of the sampling distribution is dependent on the sample size n.

As the sample size increases, the standard deviation of the sampling distribution decreases.

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The test has demonstrated a good level of test-retest reliability.

If a student took an assessment on their math ability at one time point and then the same assessment a month later, with no substantial changes such as getting a tutor, and their math ability was the same between time 1 and time 2, then the assessment has achieved Test-Retest Reliability.Test-Retest Reliability: Test-Retest reliability is the measure of consistency of a test over time. A test has test-retest reliability if a person performs similarly on the same test taken at two different times.A reliable test must always provide consistent results. Therefore, if the math ability was the same between time 1 and time 2, and no substantial changes occurred during that time, then the test has demonstrated a good level of test-retest reliability.

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Answer:

120

Step-by-step explanation:

20 percent of 600 is 120 so he will get 120

The value of x in the right triangle when calculated is approximately 13.8 units

Given the right-angled triangle

The side length x can be calculated using the following sine ratio

So, we have

sin(39) = x/22

To find x, we can use the fact that sin(39 degrees) = x/22 and solve for x.

First, we can use a calculator to find the value of sin(39 degrees), which is approximately 0.6293.

Then, we can set up the equation:

0.6293 = x/22

To solve for x, we can multiply both sides by 22:

0.6293 * 22 = x

13.8446 = x

Rewrite as

x = 13.8446

Approximate the value of x

x = 13.8

Therefore, x is approximately 13.8 in the triangle

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Answer:

  5 cm

Step-by-step explanation:

You want to know the radius of a drain pipe that empties a cylindrical tank of height 20 cm and radius 30 cm in 2 minutes when the flow rate is 6 cm/s.

The rate of emptying the cylindrical tank is its volume divided by the time it takes to empty.

  V = πr²h

  V = π(30 cm)²(20 cm) = 18000π cm³

If this volume is drained in 2 minutes = 120 seconds, the flow rate is ...

  (18000π cm³)/(120 s) = 150π cm³/s

The area of the drain pipe can be found by dividing this volumetric flow rate by the speed of the flow:

  (150π cm³/s)/(6 cm/s) = 25π cm²

The radius of the drain pipe is that of a circle with area 25π cm²:

  A = πr²

  25π cm² = πr²

  r² = 25 cm² . . . . . divide by π

  r = 5 cm

The radius of the drain pipe is 5 cm.

__

Additional comment

The 20 cm height of the tank is emptied in 120 seconds, so the rate of change of height is 20/120 = 1/6 cm/s. The exit pipe has a flow rate of 6 cm/s, which is 6/(1/6) = 36 times the rate of change in the tank.

The height change is inversely proportional to the area, which is proportional to the square of the radius. So the radius ratio is √36 = 6, meaning the drain must have a radius of (30 cm)/6 = 5 cm.

A  homogeneous linear differential equation with constant coefficients has the form a_

n y^{(n)} + a_{n-1} y^{(n-1)} + ... + a_1 y' + a_0 y = 0,

where a_n, a_{n-1}, etc. are all constants. The general solution of this equation is given by y = c_1 e^{\lambda_1 t} + c_2 e^{\lambda_2 t} + ... + c_n e^{\lambda_n t}, where c_1, c_2, etc. are constants and \lambda_1, \lambda_2, etc. are the roots of the characteristic equation a_n \lambda^n + a_{n-1} \lambda^{n-1} + ... + a_1 \lambda + a_0 = 0.

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