Question

If x represents the number of admissions counselors, how many admissionscounselors should the college employ to maximize its profit?

Answer 1

Given,

The function for the profit is,

$y=-10x^2+1500x-35000$

Differentiating the function with respect to x then,

$\begin{gathered} \frac{dy}{dx}=\frac{d}{dx}(-10x^2+1500x-35000) \\ =-20x+1500 \end{gathered}$

Check for maximum by second order differentiation,

Differentiating the function with respect to x then,

$\begin{gathered} \frac{d^2y}{dx^2}=\frac{d}{dx}(-20x^{}+1500) \\ =-20 \end{gathered}$

Negative sign shows the maximum.

For maximum, taking dy/dx=0 then,

$\begin{gathered} -20x+1500=0 \\ 1500=20x \\ x=75 \end{gathered}$

Subsituing the value of x in the function then,

$\begin{gathered} y=-10(75)^2+1500\times75-35000 \\ =-56250+112500-35000 \\ =21250 \end{gathered}$

Hence, 75 admissions counselors should the college employ to maximize its profit.