What is an inequality that describes the values of x of our box

Question

The sum of the length and must be 30 cm. The height must be 3 cm less than the length. The box should have the greatest possible volume.

Answer 1

Sol, here's how to tackle this one:

I presume the word "width" is missing after the word "and" in the first sentence.

Let the height = H

width = W

length = L

Then, H = L - 3 from the second sentence of the question.

So, L = H + 3 . . . . . . . . . . . . . (1)

And L + W = 30 from the first sentence.

=> H + 3 + W = 30 . . . . . . .from (1)

=> W = 27 - H . . . . . . . . .(2)

Volume of box, V = H x L x W

= H (H + 3) (27 - H) from (1) and (2)

= -H³ + 24H² + 81H

For maximum volume, differentiate V wrt H and equate to zero.

dV/dH = -3H² + 48H + 81

Equate this to zero: -3H² + 48H + 81 = 0

H² - 16H - 27 = 0 . . . . . . . . . dividing through by -3

This is a quadratic that doesn't have simple factors, so you need to solve it by the quadratic formula.

That will give 2 values of H, one of which will be negative and can be discarded and one positive which is the desired value of H.

I leave that as an exercise for you.