Differentiation Practical Problem with equation of motion. Maths methods

Question

So I am having a bit of trouble with this question. I get that I have to use optimisation but I am not sure how. 'find the least area of sheet metal required to make an open baking dish of square base and vertical sides capacity 2048cm cubed. I have tried to use optimisation by making the lenght times width times height formula = 2048, but I don't think that is what I should be doing. It is confusing me because there are so many unknown varliables.

Answer 1

It is open container so there are $5$ surfaces. Now say the side of the square base is $x$ and height of the container is $y$

Given volume $V = x^2y = 2048$ ...(i).

You need to minimize the surface area as you need to use least amount of metal sheet.

Total surface area $S = x^2 + 4xy$ ...(ii)

$S = x^2 + \frac{2048 \times 4}{x}$ (substituting value of $y$ from (i))

At extrema, $\frac{dS}{dx} = 2x - \frac{2048 \times 4}{x^2} = 0 \implies x^3 = 4096$

This gives you $x = 16$ (side of the square base) and from (i), you can find the height which comes to $8$.

Now you can find minimum $S$ using (ii) (you can also do second derivative test to make sure this is minima, which it is).