You are in business and have a product that sells for $10 that 1000 people buy each month. You want to make a profit and wonder whether you should raise the price in increments of a dollar. For each dollar you raise the price, you lose 100 customers. How many dollars can you go up to maximize your profit, or should you go up at all?

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FrimmyD743877 FrimmyD743877 · Answer 1 · 2022-10-27T20:05:54+02:00

Let:

P = Profit

n = Number of increments by $1

R = Revenue

The profit is given by:

[tex]\begin{gathered} P=10+1n \\ P=10+n \end{gathered}[/tex]

Therefore, the revenue is:

[tex]R=(10+n)(1000-100n)[/tex]

Expand the equation using the distributive property:

[tex]\begin{gathered} R=10000-1000n+1000n-100n^2 \\ R=-100n^2+10000 \end{gathered}[/tex]

The maximum profit of this quadratic function is located at the vertex, we can find the vertex as follows:

[tex]\begin{gathered} V(h,k) \\ h=-\frac{b}{2a} \\ a=-100 \\ b=0 \\ k=R(h)=R(0)=10000 \end{gathered}[/tex]

The vertex is (0,10000) in another words the maximum profit is achieve if you don't raise the price.