Question

A rectangular prism must have a base with an area of no more than 27 square meters. The width of the base must be 9 meters less than the height of the prism. The length of the base must be 6 meters more than the width of the base. Find the maximum height of the prism.
Let x = the height of the prism
x – 9 =

Answer 1

Let x = height. Then width = x − 9 and length = (x − 9) + 6 = x − 3. The base area constraint:
(x − 9)(x − 3) ≤ 27.

Expand and simplify:
x^2 − 12x + 27 ≤ 27 ⇒ x^2 − 12x ≤ 0 ⇒ x(x − 12) ≤ 0.

So 0 ≤ x ≤ 12. Since width = x − 9 must be positive, x > 9. Thus 9 ≤ x ≤ 12, and the maximum height is x = 12 meters.

(At x = 12, width = 3 m, length = 9 m, base area = 27 m².)