A small theater has a seating capacity of 2000.When the ticket price is $20, attendance is 1500.For each $1 decrease in price, attendance increases by 50.The revenue R of the theater as a function of ticket price x is as follows $R ( x ) = - 50 x ^ { 2 } + 2500 x , 10 \leq x \leq 20$ What ticket price will yield a maximum revenue?
A)$55.00
B)$35.00
C)$25.00
D)$45.00
E)$65.00

Question

A small theater has a seating capacity of 2000.When the ticket price is $20, attendance is 1500.For each $1 decrease in price, attendance increases by 50.The revenue R of the theater as a function of ticket price x is as follows $R ( x ) = - 50 x ^ { 2 } + 2500 x , 10 \leq x \leq 20$ What ticket price will yield a maximum revenue?
A)$55.00 
B)$35.00 
C)$25.00 
D)$45.00 
E)$65.00

Quizplus · Accepted Answer

The maximum revenue is achieved at the vertex of the parabola represented by the revenue function $R(x) = -50x^2 + 2500x$. The x-coordinate of the vertex can be found using the formula $-\frac{b}{2a}$, where $a = -50$ and $b = 2500$. Thus, the ticket price for maximum revenue is $-\frac{2500}{2(-50)} = 25$, which corresponds to choice C.

A Small Theater Has a Seating Capacity of 2000 R(x)=−50x2+2500x,10≤x≤20R ( x ) = - 50 x ^ { 2 } + 2500 x , 10 \leq x \leq 20R(x)=−50x2+2500x,10≤x≤20

A Small Theater Has a Seating Capacity of 2000 $R ( x ) = - 50 x ^ { 2 } + 2500 x , 10 \leq x \leq 20$