Question

The profit P(x) obtained by manufacturing and selling x units of a certain product is given by P(x) = 60x - x2. Determine the number of units that must be produced and sold to maximize the profit. What is the maximum profit?

Answer 1

Answer:

The number of units that must be produced and sold to maximize the profit is 30 units

$30\text{ units}$

The maximum profit is;

$\text{ \ 900}$

Explanation:

Given that the profit P(x) obtained by manufacturing and selling x units of a certain product is given by;

$P(x)=60x-x^2$

The maximum point is at;

$P^{\prime}(x)=0$

Differentiating P(x);

$\begin{gathered} P^{\prime}(x)=60-2x=0 \\ 60-2x=0 \\ 2x=60 \\ x=\frac{60}{2} \\ x=30 \end{gathered}$

The number of units that must be produced and sold to maximize the profit is 30 units

Substituting x into p(x);

$\begin{gathered} P(30)=60(30)-30^2 \\ P(30)=900 \end{gathered}$

The maximum profit is;

$\text{ \ 900}$