Question

The ticket office at Orchestra Center estimates that if it charges x dollars for box seats for a concert, it will sell 50 - x box seats. The function S = 50x - x2 gives the estimated income from the sale of box seats. Graph the function, and use the graph to find the price for box seats that will give the greatest income.

Answer 1

Here, the given function,

`S(x) = 50 x - x ^2` ----- (1)

For the maximum value of s(x),

`s'(x) = 0`

Where s'(x) is the first derivative of S(x)

By differetiating equation (1) with respect to x,

We get,

S'(x)= 50 - 2x

⇒ For maximum value of S(x),
`s'(x) = 0`

⇒ 50 - 2x = 0

⇒ 2x = 50

⇒ x = $ 25

Answer 2

To estimate the price of box seats for the ticket office to have the greatest income, we derive the equation and equate the derivative to zero.
S = 50x - x²
Deriving,
dS = 50 - 2x = 0
The value of zero from the generated equation is 25. Thus, the company should sell tickets at approximately $25.