Question

The maximum profit that a company can make if the profit function is given by
\(P(x)=32+24x-18x^2\) is:

• A. 48
• B. 40
• C. 36
• D. 42

Answer 1

The Correct Option is B

To find the maximum profit that the company can make, we need to determine the maximum value of the profit function given by \(P(x)=32+24x-18x^2\). The function is a quadratic equation in the form \(ax^2+bx+c\) where a=-18, b=24, and c=32.

Since the coefficient of \(x^2\) is negative, the parabola opens downwards, indicating that the vertex of the parabola will give us the maximum point.

The vertex form of a quadratic function is found using the formula for the x-coordinate of the vertex, \(x=-\frac{b}{2a}\).

Substituting the given values:

\(x = -\frac{24}{2 \times -18} = -\frac{24}{-36} = \frac{2}{3}\)

Now, substitute \(x = \frac{2}{3}\) back into the profit function to find the maximum profit:

\(P\left(\frac{2}{3}\right) = 32 + 24\left(\frac{2}{3}\right) - 18\left(\frac{2}{3}\right)^2\)

\( = 32 + 16 - 18 \times \frac{4}{9}\)

\( = 32 + 16 - 8\)

\( = 40\)

Therefore, the maximum profit the company can make is 40.