# Linear programming extra credit problem. Help!

1. Jul 12, 2006

### Revolver

A manufacturer of tennis rackets makes a profit of $15 on each oversized racket and$8 on each standard racket. To meet dealer demand, daily production of standard rackets should be between 30 and 80 (inclusive), and prdouction of oversized rackets should be between 10 and 30 (inclusive). To maintain high quality, the total number of rackets produced should not exceed 80 per day. How many of each type should be produced to maximize the profit?

Answer the following. Show all work.
1. Write the constraints and optimal equation.
2. Graph the region of feasible constraints.
3. Find all corner points.
4. Evaluate the optimal equation at each corner point.
5. Summarize your findings in a word statement.

2. Jul 12, 2006

### NateTG

And I'd like a pony.

You're not likely to get help from people here unless you indicate what you've attempted and failed with. Especially on things like this where there isn't anything particuarly tricky.

If you don't know what some of the words mean, then ask:

What does 'feasible region' mean?

And you might actually get a reasonable answer.

Until then, I, for one, will be waiting for my pony (and it had better be pink).

3. Jul 12, 2006

### HallsofIvy

Staff Emeritus
As NateTG said, do something yourself so we can see what kind of help you need- it wouldn't help you for us to do the problem for you (especiallly if you expect your teacher to give you "extra credit" thinking you did the work yourself!).

Get started: Assign names to "number of standard rackets produced daily" and "number of oversized rackets produced daily. Now, using those names, how would you write "daily production of standard rackets should be between 30 and 80 (inclusive)" as an inequality? How would you write "production of oversized rackets should be between 10 and 30 (inclusive)" as an inequality. How would you write "the total number of rackets produced should not exceed 80 per day" as an inequality?

How would you graph those inequalities?

How would you write the profit function from "A manufacturer of tennis rackets makes a profit of $15 on each oversized racket and$8 on each standard racket. "

4. Jul 12, 2006

### Revolver

I've already started on the problem. As I'm new, I didn't know you had guidelines as to how to post questions for problems.

I already know let X be the number of oversized rackets, and let Y be the number of standard rackets.

The inequality part is what confuses me.

30 >= X >= 80 ?

How do you graph that? The problem is due in two and a half hours so I'm probably screwed. Heh.

5. Jul 12, 2006

### NateTG

It might be less confusing if you write the inequalities down one at a time. Instead of writing:
$$30 \leq x \leq 80$$
write
$$30 \leq x$$
$$x \leq 80$$
or something similar.

6. Jul 13, 2006

### HallsofIvy

Staff Emeritus
Since 30 is not greater than 80 that can't be right- you have the inequality signs reversed.
30<= X<= 80.

The best way to graph inequalities is to first graph the equation.
Set up an X,Y graph. X= 30 is a vertical straight line. X= 80 is also a vertical straight line. 30<= X<= 80 is the strip between those lines.
Same thing for 10<= Y<= 30- it's a horizontal strip. The rectangle where they overlap gives (X,Y) that satisfies both.
The last constraint is X+ Y<= 80. X+ Y= 80 is the line through (80,0) and (0,80). Since 0+ 0< 80, X+Y<= 80 is the side of that line that contains (0,0). The "feasible region" is the place where all three sets overlap. You can get the corner points by solving the equations of the lines that intersect at that corner simultaneously.