Optimizing Restaurant Seats for Maximum Profit

[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?

 

[berkeman]

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!

 

[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.

 

[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?

 

[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.