Question

A cylinder has a volume of 300 cubic inches. The top and bottom parts of the cylinder cost $2 per square inch. And the sides of the cylinder cost $6 per square inch. What are the dimensions of the Cylinder that minimize cost based on these constraints?

Answer 1

Answer with explanation:

Volume of Cylinder=V =πr²h=300 cm³

           $h=\frac{300}{\pi r^2}$

                                                  ----------------------(1)

It is given that,  top and bottom parts of the cylinder cost $2 per square inch. And the sides of the cylinder cost $6 per square inch.

S=2πr(h+r),where r is the Radius of Cylinder,and h is the height of cylinder.

Total Cost for making the cylinder ,which minimizes the cost (T)

                      = 2 × 2πr²+ 6 ×2πr h

                   T  = 4 π r²+12 π r h

   $T=4\pi r^2+12\pi \times r \times \frac{300}{\pi r^2}\\\\T=4\pi r^2+\frac{3600}{r}$

For, Maximum  or minimum, we will differentiate the above function with respect to ,r

$\frac{dT}{dr}=8\pi r -\frac{3600}{r^2}$

$8\pi r -\frac{3600}{r^2}=0\\\\8\pi r^3=3600\\\\r^3=\frac{3600}{8\pi}\\\\r=[\frac{3600}{8\pi }]^{\frac{1}{3}}\\\\r=[\frac{450}{3.14}]^{\frac{1}{3}}\\\\r=(143.312)^{\frac{1}{3}$

r=5.24 inch

Substituting the value of ,h in equation (1)

$h=\frac{300}{3.14\times 27.3855}\\\\h=\frac{300}{85.990}\\\\h=3.49$

Dimensions of the Cylinder that minimize cost based on these constraints are,

Radius=5.24 inch

Height=3.49 inch