Question

find the volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard of dimensions 15 cm by 24 cm and then turning up the sides​

Answer 1

The answer is in the above pics.

Answer 2

Given :

Dimensions of cardboard is 15cm by 24cm.

To find :

The volume of the largest box that can be made by cutting equal squares out of the corners of a piece of cardboard.

Solution :

V = (24 - 2x)(15 - 2x)x

V = 360x - 78x² + 4x³

Volume maximizes when  dV / dx = 0

dV / dx = 360 - 156x + 12x² = 0

360 - 156x + 12x² = 0

x² - 13x + 30 = 0

(x - 10)(x - 3) = 0

x = 10 and x = 3

Putting x = 10 in V = 360x - 78x² + 4x³

V = 360x - 78x² + 4x³

V = 360×10 - 78×10² + 4×10³

V = 3600 - 7800 + 4000

V = -200

V cannot be negative.

So, x cannot be 10

Putting x = 3 in V = 360x - 78x² + 4x³

V = 360x - 78x² + 4x³

V = 360×3 - 78×3² + 4×3³

V = 1080 - 702 + 108

V = 486cm³

Therefore, the volume of the largest box that can be made by cutting equal squares out of the corners is 486cm³