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 Question from Joe, a student: A local artist sells her portraits at the Eaton Mall. Each portrait sells for $\$20$and she sells an average of 30 per week. In order to increase her revenue, she wants to raise her price. But she will lose one sale for every dollar increase in price. If expenses are$\$10$ per portrait, what price should be set to maximize the weekly profits? What is the maximum weekly profit? I NEED SOMEONE TO EXPLAIN THIS TO ME WITH FULL DETAIL AND FULL EXPLANATION. STEP BY STEP WOULD REALLY HELP! THANX!

Hi Joe,

The goal is to maximize profit and profit is income minus expenses. At the moment her income is

$20 × 30 per week and her expenses are$10 × 30 per week.

Thus her profit is

($20 × 30) - ($10 × 30) per week.

If she increases the price to $(20 + 1) her sales reduce to (30 - 1) per week and her expenses are$10 × (30 - 1) per week so her weekly profit is

[$(20 + 1) × (30 - 1)] -$10 × (30 - 1) per week.

If she increases the price to $(20 + 2) her sales reduce to (30 - 2) per week and her expenses are$10 × (30 - 2) per week so her weekly profit is

[$(20 + 2) × (30 - 2)] -$10 × (30 - 2) per week.

Now I can see the pattern. If she increases the price to $(20 + x) her sales reduce to (30 - x) per week and her expenses are$10 × (30 - x) per week so her weekly profit is

[$(20 + x) × (30 - x)] -$10 × (30 - x) per week.

This is the profit function so you can use the calculus you know to maximize this function.

Write back if you need more assistance,
Penny

Hi Joe.

If you don't know calculus, you can still solve the problem.

When you set the expression Penny wrote equal to "y" and multiply it out, you get a quadratic function. This means a parabola. You'll find, if you check, that this parabola opens downward. That means that the
vertex of the parabola is at the topmost point.

If you write the quadratic function in vertex form, you can answer the question by just reading off the y value of the vertex. See
http://mathcentral.uregina.ca/QQ/database/QQ.09.07/s/mark2.html for a
worked example.

Cheers,
Stephen La Rocque

Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.