Question

Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 23 ft by 13 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.

Answer 1

Answer:

Step-by-step explanation:

Given that rectangle  has side length 23x13 feet.

When squares of side x are cut from all the four sides we have dimensions as

23-2x and 13-2x and height x

Volume $=x(23-2x)(13-2x)$

$=x(299-72x+4x^2)\\=299x-72x^2+4x^3$

Use derivative test

First derivative V'(x) $V'(x)= 299-144x+12x^2\\V"(x) = -144+24x$

Equate I derivative to 0

We get approximate value of x = 2.671, 9.329

Since width is 13 feet it cannot be 9.329

Hence x = 2.671

V"(x) = -80 <0

So maximum volume is

$299x-72x^2+4x^3\\=299(2.671)-72(2.671)^2 +4(2.671)^3\\\\=361.186$