Question

The total revenue for Dante's Villas is given as the function R(x)=700x−0.5x^2, where x is the number of villas rented. What number of villas rented produces the maximum revenue?

Answer 1

The revenue function is,

$R(x)=700x-0.5x^2$

For maximum and minimum value, the first derivative of function is equal to 0.

Determine the first derivative of revenue function.

$\begin{gathered} \frac{d}{dx}R(x)=\frac{d}{dx}(700x-0.5x^2) \\ R^{\prime}(x)=700-0.5\cdot2x \\ =700-x \end{gathered}$

For maximum and minimum value, R'(x) = 0. So

$\begin{gathered} 0=700-x \\ 700=x \end{gathered}$

Since x = 700 corresponds to maximum or minimum value of revenue.

Determine the second derivative of revenue function.

$\begin{gathered} \frac{d}{dx}R^{\prime}(x)=\frac{d}{dx}(700-x) \\ R^{\doubleprime}(x)=-1 \end{gathered}$

Since second derivative of revenue function is less than 0 for every value of x. So x = 700 corresponds to maximum value of revenue.

So 700 villas rented to produce maximum revenue.

Answer: 700 villas