Q. 55 - Calculus

Question

Your company produces cylindrical metal oil drums that must each hold $40$ cubic feet of oil. How should the oil drums be constructed so that they use as little metal as possible? Can they be constructed to use as much metal as possible? 

Step-by-Step Solution

Answer

Ans:  To get minimum surface area, the oil drums should be constructed as $3.0767$ feet high with a radius of $2.335$ feet. There is no global maximum.

Step 1. Given information.

given,

     The cylindrical metal oil drum holds $40$ cubic feet of oil.

Step 2. The objective is to find how should oil drums can be constructed so that they use as little metal as possible.

The surface area of the cylinder is as follows.

    $S=2{\mathrm{\pi r}}^2+2\mathrm{\pi rh}$

The volume of the box is as follows.

    $$\begin{gathered} V={\mathrm{\pi r}}^2\mathrm{h} \\ 40={\mathrm{\pi r}}^2\mathrm{h} \\ \mathrm{h}=\frac{40}{{\mathrm{\pi r}}^2} \end{gathered}$$

Step 3. The derivative of the surface area is calculated below.

     $\begin{array}{r}S=2\pi r^2+2\pi r\left(\frac{40}{\pi r^2}\right)\\ S=2\pi r^2+\frac{80}{r}\\ S^{\mathrm{\prime }}=4\pi r-\frac{80}{r^2}0\\ \\ 4\pi r^3-80=0 \\ 4\pi r^3=80 \\ \pi r^3=20 \\ r=\sqrt[3]{\frac{20}{\pi }}\\ =1.8533\end{array}$

Step 4. The height of the cylinder is calculated below.

  $\begin{array}{r}h=\frac{40}{\pi r^2} \\ =\frac{40}{\pi \times (1.853{)}^2}\\ =3.0767\end{array}$

Therefore, To get minimum surface area, the oil drums should be constructed as $3.0767$ feet high with a radius of $2.335$ feet.

There is no global maximum.