Solve the problem.
-The stadium vending company finds that sales of hot dogs average 50,000 hot dogs per game when the hot dogs sell for $\$ 2.50$ each. For each 50 cent increase in the price, the sales per game drop by 5000 hot dogs. What price per hot dog should the vending company charge to realize the maximum revenue?
A) $\$ 4.00$
B) $\$ 2.50$
C) $\$ 5.00$
D) $\$ 3.75$

Question

Quizplus · Accepted Answer

To maximize revenue, we need to find the price per hot dog that results in the highest product of price and quantity sold. Let $x$ represent the number of 50 cent increases in the price. The price per hot dog can be represented as $2.50 + 0.50x$ dollars, and the quantity sold can be represented as $50000 - 5000x$ hot dogs. The revenue function, $R$, is the product of the price and quantity, so $R(x) = (2.50 + 0.50x)(50000 - 5000x)$. Simplifying, we get $R(x) = 125000 - 12500x + 25000x - 2500x^2$, or $R(x) = -2500x^2 + 12500x + 125000$. To find the maximum revenue, we need the vertex of this parabola, which occurs at $x = -b/2a$. Here, $a = -2500$ and $b = 12500$, so $x = -12500/(2*(-2500)) = 2.5$. This means the price increase should be $2.5 * 0.50 = \$1.25$, making the optimal price $2.50 + 1.25 = \$3.75$.

Solve the Problem $2.50\$ 2.50$2.50 Each for Each 50 Cent Increase in the Price, the of Hot

Solve the Problem $\$ 2.50$ Each for Each 50 Cent Increase in the Price, the of Hot