Question

A box without a top is made from a rectangular piece of cardboard with dimensions 12 cm by 10 cm, by cutting out square corners with side length x.
what x-value gives the greatest volume?
use technology to estimate your answer to the nearest tenth.

Answer 1

Answer:

  x ≈ 1.8 cm gives the greatest volume

Step-by-step explanation:

After cutting x cm from each corner in each direction, the cardboard can be folded up to make a box that is x cm deep and (12 -2x) by (10 -2x) in length and width. Clearly, values of x are limited to 5 or less, since cutting 5 cm from each side would leave a width of zero. Then the volume is given by ...

  V = x(12 -2x)(10 -2x)

The plot below shows the value of this cubic equation for volume, and identifies the peak as (x, V) ≈ (1.8, 96.8). That is, a cut of 1.8 cm will result in a box of approximate volume 96.8 cm³.