$MathMaster logo$

MathMaster

Home
general
marginal income when a hairdresser sets a fee of 4 per haircut she notices that the number of clients she serves in a week

Question

(Marginal income) When a hairdresser sets a fee of $4 per haircut, she notices that the number of clients she serves in a week is 100, on average. By raising the rate to $5, the number of customers per week drops to 80. Assuming a linear demand equation between the price and the number of customers, determine the marginal revenue function. Then find the price that produces zero marginal revenue.

257

likes

1287 views

Answer to a math question (Marginal income) When a hairdresser sets a fee of $4 per haircut, she notices that the number of clients she serves in a week is 100, on average. By raising the rate to $5, the number of customers per week drops to 80. Assuming a linear demand equation between the price and the number of customers, determine the marginal revenue function. Then find the price that produces zero marginal revenue.

$Expert avatar$

4.9

87 Answers

1. \text{Calculate the slope } m:

m = \frac{Q_2 - Q_1}{P_2 - P_1} = \frac{80 - 100}{5 - 4} = \frac{-20}{1} = -20

2. \text{Calculate the intercept } b \text{ using the point } (4, 100):

100 = -20 \cdot 4 + b

100 = -80 + b

b = 180

\text{Thus, the demand equation is:}

Q = -20P + 180

3. \text{Substitute the demand equation into the revenue function:}

R = P \cdot Q

R = P \cdot (-20P + 180)

R = -20P^2 + 180P

4. \text{Find the marginal revenue } MR \text{ by taking the derivative of } R:

MR = \frac{dR}{dP} = \frac{d}{dP} (-20P^2 + 180P)

MR = -40P + 180

5. \text{Set } MR = 0 \text{ and solve for } P:

-40P + 180 = 0

40P = 180

P = \frac{180}{40}

P = 4.5

\text{The price that produces zero marginal revenue is } \$4.50

Frequently asked questions (FAQs)

Question: Expand and factorize the expression 3x² - 6x - 21 using the distributive property.

+

What is the value of x when the exponential function f(x) = 10^x is equal to e^x?

+

Math question: Given a right triangle with one leg measuring 8 units and the hypotenuse measuring 17 units, what is the measure of the other leg?

+

New questions in Mathematics

calculate the derivative by the limit definition: f(x) = 6x^3 + 2

1. Suppose we have a good whose quantity supplied changed from 100 to 120 units when the price increased from $10 to $12 per unit. Compute the price elasticity of supply using the midpoint method

An integer is taken at random from the first 40 positive integers. What is the probability that the integer is divisible by 5 or 6?

Solve this mathematical problem if 3/5 of a roll of tape measures 2m. How long is the complete roll?

Emma is on a 50 m high bridge and sees two boats anchored below. From her position, boat A has a bearing of 230° and boat B has a bearing of 120°. Emma estimates the angles of depression to be about 38° for boat A and 35° for boat B. How far apart are the boats to the nearest meter?

4x + 8y = 5 2x + 4y = 10

In the telephone exchange of a certain university, calls come in at a rate of 5 every 2 minutes. Assuming a Poisson distribution, the average number of calls per second is: a) 1/8 b) 1/12 c) 1/10 d) 2/5 e) 1/24

A storage maker price is $2.50 per square feet. Find the price of a custom shed 4 yards long, and 5yards wide and 8 feet tall

A function is considered exponential when it has a base with positive values greater than zero and different from one, where the exponent is an unknown. An important characteristic of exponential functions is that they show rapid growth or decay as an independent variable increases or decreases. Given the function 25^(x+3)=125, it is calculated that x has the value of

9 x² + 2x + 1 = 0

Show work on 4108 divided by 4

For what values of m is point P (m, 1 - 2m) in the 2⁰ quadrant?

How many cards do you expect to pull from a poker deck until you get an ACE?

Gender and communication : Answer the question ( 1 paragraph is ok) . Please can you write about women? Compared to your other identities, how much of a role does gender play in your life? And has your own sex/gender offered you privileges or disadvantages? How so?

Let I be an interval and let f : I → R be a continuous function such that f(I) ⊂ Q. Show (in symbols) that f is constant.