Question

A company plans to manufacture a rectangular bin with a square base, an open top, and a volume of 13,500 in3. Determine the dimensions of the bin that will minimize the surface area. What is the minimum surface area

Answer 1

Answer:

Dimensions 30 in x 30 in x 15 in

Surface Area = 2,700 in²

Step-by-step explanation:

Let 'r' be the length of the side of the square base, and 'h' be the height of the bin. The volume is given by:

$V=13,500=h*r^2\\h=\frac{13,500}{r^2}$

The total surface area is given by:

$A=4*hr+r^2$

Rewriting the surface area function as a function of 'r':

$A=4*\frac{13,500}{r^2} *r+r^2\\A=\frac{54,000}{r}+r^2$

The value of 'r' for which the derivate of the surface area function is zero, is the length for which the area is minimized:

$A=54,000*r^{-1}+r^2\\\frac{dA}{dr}=0= -54,000*r^{-2}+2r\\\frac{54,000}{r^2}=2r\\ r=\sqrt[3]{27,000}\\r=30\ in$

The value of 'h' is:

$h=\frac{13,500}{30^2}\\ h=15\ in$

The dimensions that will ensure the minimum surface area are 30 in x 30 in x 15 in.

The surface area is:

$A=4*15*30+30^2\\A=2,700\ in^2$