Question

A company finds that it can make a profit of P dollars each month by selling a patterns, according to theformulaP(x)=-0.002² +3.5x - 800.How many patterns must it sell each month to have a maximum profit?____ patternsWhat is the maximum profit? $____

Answer 1

To determine the maximum profit we notice that this is a quadratic function with negative leading term which means that its maximum is its vertex; if we complete the square we can find both the answers we are looking for so let's complete the squares:

$\begin{gathered} P(x)=-0.002x^2+3.5x-800 \\ =-0.002(x^2-1750x)-800 \\ =-0.002(x^2-1750x+(-\frac{1750}{2})^2)-800+0.002(-\frac{1750}{2})^2 \\ =-0.002(x-875)^2+731.25 \end{gathered}$

Hence, we can write the function as:

$P(x)=-0.002(x-875)^2+731.25$

From it we notice that the vertex of the function is (875,731.25) and therefore:

• If the company sells 875 patterns the have a maximum profit.

,

• The maximum profit is $731.25