A stadium has 64000 seats when the ticket is $10, there is an attendance of 25000. when the...

Question:

A stadium has 64000 seats when the ticket is $10, there is an attendance of 25000. when the ticket is $7 there is an attendance of 32000. find the highest revenue?

Maximizing a Function

A function is maximized at a local maxima or global maxima whenever the first derivative is equated to zero, and for the value it is zero, the second derivative is negative.

Answer and Explanation:

Let us find the slope of the demand function if the slope is linear:

$$\begin{align} Slope = \left( \dfrac{32000 - 25000}{7 - 10} \right) = \left( \dfrac{-7000}{3} \right) \end{align}\\ $$

Thus, the equation of demand line is:

$$\begin{align} y - 32000 = (x - 7) \left( \dfrac{-7000}{3} \right) \\[0.3cm] 3(y - 32000) = -7000x + 49000 \\[0.3cm] 3y = - 7000x + 49000 + 96000 \\[0.3cm] y = \left( \dfrac{-7000x + 145000}{3} \right) \\[0.3cm] \end{align}\\ $$

The revenue function can then be written as:

$$\begin{align} R(x) &= xy \\[0.3cm] &= x \left( \dfrac{-7000x + 145000}{3} \right) \\[0.3cm] &= \left( \dfrac{-7000x^2 + 145000x}{3} \right) \\[0.3cm] \end{align}\\ $$

Hence, the derivative of revenue function and its equation with 0 is:

$$\begin{align} R'(x) = \left( \dfrac{-14000x + 145000}{3} \right) = 0 \\[0.3cm] \implies x = \left( \dfrac{145}{14} \right) \\[0.3cm] R''(x) = \left( \dfrac{-14000}{3} \right) \leq 0 \end{align}\\ $$

Thus, maximum revenue is:

$$\begin{align} R \left( \dfrac{145}{14} \right) &= \left( \dfrac{-7000 \left( \dfrac{145}{14} \right)^2 + 145000 \left( \dfrac{145}{14} \right)}{3} \right) \\[0.3cm] &= \left( \dfrac{5256250}{21} \right) \\[0.3cm] &= 250297\left( \dfrac{13}{21} \right) \\[0.3cm] \end{align}\\ $$

Thus, the maximum revenue is roughly $250,297.


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