How to calculate the smallest surface needed to have a cylinder of 1.75 dm^3?

Question

I'm new to optimization, and I'm doing simple exercises. Nevertheless I have the impression that, in order to find the smallest surface needed to get a $1.75$ dm^3 cylinder, I should look for the minima of a function $f_s(r,h)= 2\pi r h + 2\pi r^2$ such that $\pi r^2h=1.75$, which implies going MUCH further than my current level by working on a bivariate function.

What is the simplest way of solving this problem?

Thanks in advance for your answer.

Answer 1

HINT

We have

• $\pi r^2h=1.75 \implies \pi rh=\frac{1.75}r$

and then including the bases

• $f_s(r)= 2\pi r(r + h)=2\pi r^2+\frac{3.5}r$

Answer 2

$$\pi r^2 h = 1.75$$

$$h = \frac{1.75}{\pi r^2}$$

What you want to optimize is

$$2\pi r (h+r) = 2\pi r \cdot \left(\frac{1.75}{\pi r^2}+r\right)=\frac{3.5}{r}+2\pi r^2$$ subject to $r >0$.

Now the question is single variable and hopefully you can take it from here.

Answer 3

It is given the Surface of this cylinder: $$A=2\pi r^2+2\pi rh$$ and the volume $$V=7/4=\pi r^2h$$. Solve this equation for $$h$$ and plugg it in the Surface Formula, which containes then only one variable.