question about optimization model

Question

It seems pretty odd to change one of the variable when trying to do some optimization.

Answer 1

You want your volume to be $33$, hence you should not let it be an inequality. when we see $V$, we should directly replace it with $33$.

You do not want to minimize the gradient of the objective function, you want to minimize the objective function.

That is you want to minimize

$$\min _r\frac{66}r+2\pi r^2$$

$$s.t. r > 0$$