# Homework Help: Max Profit

1. Jun 25, 2006

### coolant

In planning a resturant, it is estimated that a profit of \$5 per seat will be made if the number of seats is between 60 and 80, inclusive. On the other hand, the profit on each seat will decrease by 5 cents for each seat above 80?

a) Find the number of seats that will produce the max profit
b) what is the max profit?

2. Jun 26, 2006

### Staff: Mentor

Make a hand plot first. What is the answer? Then use calculus to find the maximum. What is the answer? They should match, eh? Welcome to PF!

3. Jun 26, 2006

### 0rthodontist

You don't really need calculus since the marginal profit is almost given to you. Just stop adding seats when the marginal profit is less than 0.

4. Jun 26, 2006

### HallsofIvy

You don't say what the "profit" is, if any, if the number of seats is less than 60 so I will assume it is not defined for x< 60.

Okay, the profit function, for x seats, is
P(x)= 5x if 60<= x<= 80
= (5- 0.05(x-80))x if x> 80

As Orthodontist said, you don't really need calculus since the marginal profit is "almost given to you" but you can:

P'(x)= 5 if 60<= x<= 80
= -0.05x+ (5- 0.05(x-80))

For what value of x is that equal to 0?

5. Jun 26, 2006

### 0rthodontist

Actually, I think the additional profit on placing the x'th seat for x > 80 is
-.05(x-1)+(5-.05(x-80))
Since with each seat, you
1. Lose 5 cents on each seat already placed, which is x-1 seats
2. Gain (5-.05(x-80)) dollars on the current seat
A strict derivative would only be approximate since this is discrete.

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