word problem (Calculus)?

Question

I keep trying these (there's 1 more) but I'm hitting a brick wall. Can anyone help me please?

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 $0.05 for each seat above 80.

a) Find the number of seats that will produce the maximum profit.

b) What is the maximum profit?

Answer 1

The profit on each seat is 5 - 0.05(x-80), for x >= 80.
So the total profit is P(x) = x(5 - 0.05(x-80))
= 5x - 0.05x^2 + 4x
= 9x - 0.05x^2

P'(x) = 9 - 0.1x, P"(x) = -0.1, so the maximum profit will occur when 0.1x = 9, i.e. at 90 seats. (We don't really need calc for this; it's just a quadratic, so we already know it has a maximum at -9/2(-0.05) = 90.)

P(90) = 90 (5 - 0.05(10)) = 90 (4.5) = $405.

[Note that spidermilk's answer of 130 is incorrect: P(130) = 130(5 - 0.05(50)) = 130 (2.5) = $325.]

Answer 2

I have never taken calculus before but I did this the best I could using trial and error, the computers calculater and basic logic.

Each seat is worth $5.00 but goes down in worth by $0.05 for every seat added after 80.

Bassicaly alls you have to do is balance out the number of seats, the amount each seat is worth and the seat profit.

SW = Seat Worth
SN = Seat Amount
SP = Seat Profit

SW ::: SA ::: SP
$5.00 : 80 : $400.00
$4.95 : 81 : $400.95
$4.90 : 82 : $401.80
$4.85 : 83 : $402.55
$4.80 : 84 : $403.20
$4.75 : 85 : $403.75
$4.70 : 86 : $404.20
$4.65 : 87 : $404.55
$4.60 : 88 : $404.80
$4.55 : 89 : $404.95
------------------------------
$4.50 : 90 : $405.00 : : : : ANSWER : : : :
------------------------------
$4.45 : 91 : $404.95
$4.40 : 92 : $404.80
(ETC.)

The maximum seat profit is $405.00
The number of seats is 90.

This should be the right answer, but I suggest you check my work just incase. I am only in grade 9 but I still hope this is helpful.

Answer 3

Okay so first you make an equation, then you find the maximum. So, we know the max is somewhere above 80.. lets make an equation:

5(x) -.5(x-80)^2 = y

so I guess that can be simplified:

13x -320 -.05x^2 = y
Derivative = 0

-.1x + 13 = 0
x = 130

that might be wrong because i didn't check math and i took calc was a while ago

oh yea, max profit =

5(130) -.5(130-80)^2 = 525