Question

A box with a square base and open top must have a volume of 13,500 cm3. Find the dimensions of the box that minimize the amount of material used. sides of base cm height cm

Answer 1

Answer:

base: 30 cm

height: 15cm

Step-by-step explanation:

Let 'b' be the length of the side of the base, and 'h' the height of the box. The volume is given by:

$V=13,500=b^2*h$

The total area of material used is:

$A=b^2+4*(b*h)$

Using both equations to solve for 'b':

$h=\frac{13,500}{b^2}\\A=b^2+4*(b*\frac{13,500}{b^2})\\A=b^2+\frac{54,000}{b}\\\frac{dA}{db}=0=2b+\frac{54,000}{b^2}\\2b^3=54,000\\b=30\ cm$

The value of 'b' for which the derivate of the area function (A) equals zero, is the value that yields the minimum area.

If b=30, the height 'h' is:

$13,500=30^2*h\\h = 15\ cm$