Given cylinders of equal surface area, fnd the one with largest Volume.

Attempt:

\(\displaystyle V = \pi r^2h\)

\(\displaystyle A = 2\pi r^2 + 2 \pi rh\)

Now I want to use Lagrange multiplier and use the area as a constraint. However, the pi is scaring me. Pi is a constant and so far I have only learnt doing it with variables. I thought of substituting pi with x for example but that is variable. Or should I just treat it as constant and go on with it like this:

\(\displaystyle 2rh = 4r\) .......................... I

\(\displaystyle r^2 = 2r\) ........................... II

\(\displaystyle A = 2\pi r^2 + 2 \pi rh\) ........................... III

From equation I, h = 2r.

Substituting in III, we get \(\displaystyle 4\pi r^3 + 4\pi r^2\), which is the same as \(\displaystyle \pi r^3 + \pi r^2\)

Now I am stuck here.

Attempt:

\(\displaystyle V = \pi r^2h\)

\(\displaystyle A = 2\pi r^2 + 2 \pi rh\)

Now I want to use Lagrange multiplier and use the area as a constraint. However, the pi is scaring me. Pi is a constant and so far I have only learnt doing it with variables. I thought of substituting pi with x for example but that is variable. Or should I just treat it as constant and go on with it like this:

\(\displaystyle 2rh = 4r\) .......................... I

\(\displaystyle r^2 = 2r\) ........................... II

\(\displaystyle A = 2\pi r^2 + 2 \pi rh\) ........................... III

From equation I, h = 2r.

Substituting in III, we get \(\displaystyle 4\pi r^3 + 4\pi r^2\), which is the same as \(\displaystyle \pi r^3 + \pi r^2\)

Now I am stuck here.

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