Question

A candy box is made from a piece of cardboard that measures 23 by 13 inches. Squares of equal size will be cut out of each corner. The sides will then be folded up to form a rectangular box. What size square should be cut from each corner to obtain maximum​ volume?

Answer 1

Answer:

2.67 inches.

Step-by-step explanation:

Assuming that we represent the size of the squares with the letter y, such that after the squares are being cut from each corner, the rectangular length of the box that is formed can now be ( 23 - 2y), the width to be (13 - 2y) and the height be (x).

The formula for a rectangular box = L × B × W

= (23 -2y)(13-2y) (y)

= (299 - 46y - 26y + 4y²)y

= 299y - 72y² + 4y³

Now for the maximum volume:

dV/dy = 0

This implies that:

299y - 72y² + 4y³ = 299 - 144y + 12y² = 0

By using the quadratic formula; we have :

$= \dfrac{-b \pm \sqrt{b^2 -4ac}}{2a}$

where;

a = 12; b = -144 and c = 299

$= \dfrac{-(-144) \pm \sqrt{(-144)^2 -4(12)(299)}}{2(12)}$

$= \dfrac{144 \pm \sqrt{20736 -14352}}{24}$

$= \dfrac{144 \pm \sqrt{6384}}{24}$

$= \dfrac{144 \pm79.90}{24}$

$= \dfrac{144 + 79.90}{24} \ \ OR \ \ \dfrac{144 - 79.90}{24}$

$= 9.33 \ \ OR \ \ 2.67$

Since the width is 13 inches., it can't be possible for the size of the square to be cut to be 9.33

Thus, the size of the square to be cut out from each corner to obtain the maximum volume is 2.67 inches.