The sum of three positive number is 30. The first plus twice the second plus three times the third add up to 60. Select the numbers so that the product of all three is as large as possilbe.

Answer 1

just write down the math for the words:

x+y+z = 30
x+2y+3z = 60

Think of the product as the volume of a rectangular solid. The solid has maximum volume when it is a cube. SO, we want x,y,z to be nearly equal in value.

1st guess: x=y=z=10
That also fits the 2nd equation.

So, xyz has a maximum value of 1000.

To check that, suppose we adjust x and y by an amount a. Then we have

(10-a)(10+a)(10)
= 10(100-a^2)
= 1000 - 10a^2
which is less than 1000