Question

. Cut out squares of the same size from the four corners and turn up the sides.
What are the dimensions of your cardboard? What is the size of the square to be cut out so that your cake can fit in your box? What is the volume of your box?
If the side of the square that you cut from the four corners of your cardboard or carton is ‘x’, what will be your equation? How can you find the measure of the square?

Answer 1

Answer:

Let's assume that, a square piece of X cm length is cut from each corner of a square cardboard, whose side length is 72 cm.

When resulting flaps are folded, base of the box will be square with side length as (72 — 2X) cm. and height will be X cm.

So the volume of the box will be =>

V = X(72 — 2X)^2 => V = 5184X — 288X^2 + 4X^3

Derivating the function,

V' = 5184 — 576X + 12X^2

To find critical points,

X^2 — 48X + 432 = 0, (X — 36)(X — 12) = 0

X = 36, X = 12

Taking second derivative, V'' = —576 + 24X

Substituting 36 and 12 in the second derivative,

V" = —576 + 24(36) = + 288

V" = —576 + 24(12) = — 288

So, when a square pieces of 12X12 cm size are cut from corners of cardboard of 72X72 cm size, box made by folding flaps generates maximum volume.

V = (72–12–12)^2×12 = 27648 cm^3