# Thread: Calculus Word Problem

1. ## Calculus Word Problem

A retailer who sells television recorders finds that she can sell 50 recorders per month when the price is $400 per unit. For each$5 decrease in price, one additional recorder can be sold. If she purchases each unit for $100 and pays$20 to a technician to adjust and fine-tune each unit when it is delivered, what level of sales will maximize monthly profit?

I've been stuck on this problem for a while and I am stumped. I would appreciate any help on it. Thanks.

2. Originally Posted by BiGpO6790
A retailer who sells television recorders finds that she can sell 50 recorders per month when the price is $400 per unit. For each$5 decrease in price, one additional recorder can be sold. If she purchases each unit for $100 and pays$20 to a technician to adjust and fine-tune each unit when it is delivered, what level of sales will maximize monthly profit?
Use what you learned back in algebra for this sort of quadratic word problems:

[html]+-----------+--------------+-----------+--------------------+-----------+
| number of | resulting | number | total | total |
| increases | price | sold | income | costs |
+-----------+--------------+-----------+--------------------+-----------+
| 0 | $400 | 50 | (50)(400) | (50)(120) | +-----------+--------------+-----------+--------------------+-----------+ | 1 |$400 - 1($5) | 50 + 1(1) | (50 + 1)(400 - 5) | (51)(120) | +-----------+--------------+-----------+--------------------+-----------+ | 2 |$400 - 2(\$5) | 50 + 2(1) | (50 + 2)(400 - 10) | (52)(120) |
+-----------+--------------+-----------+--------------------+-----------+
| ... | | | | |
+-----------+--------------+-----------+--------------------+-----------+
| x | | | | |
+-----------+--------------+-----------+--------------------+-----------+[/html]
Keep going (in the "..." part) until you see the pattern. Then plug "x" into that pattern to create your "total income" and "total costs" expressions. Subtract to find the "profit" expression.

Then use what you learned back in algebra about graphing quadratics (specifically, that the max/min point is given by the vertex) to find the solution.

If you get stuck, please reply showing how far you have gotten. Thank you!