Math question - maximizing profit?

Question

Mr. A is selling eggs and Mr. B wants to buy eggs.

If Mr. A sells the eggs for 56 cents each, then Mr. B will buy 80 of them.

Mr. A wants to raise the price of the eggs, but for every cent that he raises his price, the number of eggs Mr. B is willing to buy will drop by one.

For example if Mr. A raises the price to 57 cents each, then Mr. B will buy 79 eggs.
If Mr. A raises the price to 58 cents each, then Mr. B will buy only 78 eggs.

In order to get the most money from Mr. B, what price should Mr. A set for his eggs?

Don't forget to explain how you found the answer!

Answer 1

well at the original price, his profit is 56 x 80 = 4480

so as price = 56 + x, then sales = 80 - x so profit = (56 + x) X (80 - x) = P = -x^2 + 80x - 56x + 4480 = -x^2 + 24x + 4480... so you want to find the vertex of the parabola.. use the formula -B/2A = -24/2(-1) = 12


or take the derivative of -x^2 + 24x + 4480 = -2x+24 set =0, then -24/-2=12

so if x = 12, then price is 56+12 = 68, and he purchases 80-12=68, then profit = 46.24

Answer 2

Hi there,
its easy to get on excel and have 56 to 80 on a columns and 80 to 56 on the other.. and find out which has the maximum product value

Let me tell you a short cut - the product of no of eggs to numbers is 56*80= 4480
The maximum profit will come when the two numbers are the same (which is 68 * 68) = 4624

This is a law of maths where in the max area of a rectangle is when it is a square; which means the two sides have to be equal.

You can also have a derivative equation to give you the same answer..
cheers!

Answer 3

56 cents X 80 eggs = $44.80
57 cents X 79 eggs = $45.03
58 cents X 78 eggs = $45.24
59 cents X 77 eggs = $45.43
60 cents X 76 eggs = $45.60
61 cents X 75 eggs = $45.75
62 cents X 74 eggs = $45.88
63 cents X 73 eggs = $45.99
64 cents X 72 eggs = $46.08
65 cents X 71 eggs = $46.15
66 cents X 70 eggs = $46.20
67 cents X 69 eggs = $46.23
68 cents X 68 eggs = $46.24 <--------
69 cents X 67 eggs = $46.23
70 cents X 66 eggs = $46.20
71 cents X 65 eggs = $46.15