(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.

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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.

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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

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