When the price of a glass of lemonade at a lemonade stand was $1.75, 400 glasses were sold. When the price was lowered to $1.50, 500 glasses were sold. Assume that the demand function is linear and that the marginal and fixed costs are $0.10 and $ 25, respectively. Find the marginal profit when 300 glasses of lemonade are sold and when 700 glasses of lemonade are sold.
A) $P ^ { \prime } ( 300 ) = 1.15 , P ^ { \prime } ( 700 ) = - 0.85$
B) $P ^ { \prime } ( 300 ) = - 0.85 , P ^ { \prime } ( 700 ) = 1.15$
C) $P ^ { \prime } ( 300 ) = 1.15 , P ^ { \prime } ( 700 ) = 0.85$
D) $P ^ { \prime } ( 300 ) = 0.85 , P ^ { \prime } ( 700 ) = - 1.15$
E) $P ^ { \prime } ( 300 ) = - 1.15 , P ^ { \prime } ( 700 ) = - 0.85$

Question

Quizplus · Accepted Answer

To find the marginal profit, we first need to determine the demand function and then the revenue and profit functions. Given two points on the demand curve, (400, 1.75) and (500, 1.50), we can find the slope of the demand function: $\frac{1.50 - 1.75}{500 - 400} = -\frac{0.25}{100} = -0.0025$. The demand function has the form $p = mx + b$, where $m = -0.0025$. Using one of the points to solve for $b$, $1.75 = -0.0025(400) + b$, we find $b = 2.75$. Thus, the demand function is $p = -0.0025x + 2.75$.The revenue function $R(x) = xp = x(-0.0025x + 2.75)$ and the profit function $P(x) = R(x) - C(x)$, where $C(x) = 0.10x + 25$ (cost function). Therefore, $P(x) = -0.0025x^2 + 2.65x - 25$.The marginal profit function is the derivative of the profit function, $P'(x) = -0.005x + 2.65$.For $x = 300$, $P'(300) = -0.005(300) + 2.65 = -1.5 + 2.65 = 1.15$.For $x = 700$, $P'(700) = -0.005(700) + 2.65 = -3.5 + 2.65 = -0.85$.Thus, the marginal profit when 300 glasses are sold is $1.15, and when 700 glasses are sold, it is $-0.85.

When the Price of a Glass of Lemonade at a Lemonade