Answer to Question #1425 in Calculus for Lauren

Question

You are planning to make an open rectangular box from a 12 by 14 cm piece of cardboard by cutting congruent squares from the corners and folding up the sides.

A) what are the dimensions of the box of largest volume you can make this way?

B) what is the volume?

Answer 1

Denote the side of the square as x.
The volume of the box would be V = x*(12-2x)*(14-2x) = 4x³ - 52x² + 168x
Let's find the roots of first derivation dV/dx:
V' = 12x2 - 104x + 168 = 0: x1 = 2 (13-2√37)=1.67 and x2 = 2(13 + 2√37)= 50.33 > 14
Thus the dimentions of the box would be:
{8.66, 10.66, 1.67}.