# A cylindrical can is to be made to hold 1000cm^3 of oil. How do you find the dimensions that will minimize the cost of metal to manufacture the can?

Jun 29, 2016

Height = 10.84 cm, radius of the base = 5.42 cm and material for the surface =37 square cm, nearly

#### Explanation:

Let height = h and radius of the base = r.

Then, volume $V = \pi {r}^{2} h = 1000$ cc

and surface area of the can

$S = 2 \pi {r}^{2} + 2 \pi r h$

Now, eliminating h, $S = S \left(r\right) = 2 \pi \left({r}^{2} + \frac{1000}{\pi {r}^{2}}\right)$

$S ' = 2 \pi \left(2 r - \frac{1000}{\pi {r}^{3}}\right) = 0$, when

$r = {\left(\frac{1000}{2 \pi}\right)}^{\frac{1}{3}} = 5.42$ cm, nearly.

Correspondingly, h= 10.84 cm, nearly

There is no maximum for S. Also,

$S ' ' = 2 \pi \left(2 + \frac{3000}{\pi {r}^{4}}\right) > 0$..

For this r= 5.42 cm, S = 37.1 $c {m}^{2}$