Question

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?

Answer 1

Let:

P = Profit

n = Number of increments by $1

R = Revenue

The profit is given by:

`\begin{gathered} P=10+1n \\ P=10+n \end{gathered}`

Therefore, the revenue is:

`R=(10+n)(1000-100n)`

Expand the equation using the distributive property:

`\begin{gathered} R=10000-1000n+1000n-100n^2 \\ R=-100n^2+10000 \end{gathered}`

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

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

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