Hello, Super Mallow!

Here's some help with #1. . .

1) A right triangle whose hypotenuse is $\displaystyle \sqrt{3}$ m long is revolved about one

of its legs to generate a right circular cone. Find the radius, height, and

volume of the cone of the greatest volume that can be made this way Code:

. . . - - - - - - *
. . . - - - - - * |
. . . - - √3 * |
. . . - - - * |√3sinθ
. . . - - * |
. . . - * θ |
* - - - - - *
√3cosθ

This triangle will be revolved about its vertical side.

The volume of a cone is: .$\displaystyle V \;=\;\frac{\pi}{3}r^2h$

Let $\displaystyle \theta$ be the left acute angle.

Then: .$\displaystyle r \,=\,\sqrt{3}\cos\theta,\;h\,=\,\sqrt{3}\sin\theta$

We have: .$\displaystyle V \;=\;\frac{\pi}{3}\left(\sqrt{3}\cos\theta\right)^ 2\left(\sqrt{3}\sin\theta\right) \;=\;\sqrt{3}\pi\sin\theta\cos^2\!\theta $

Can you finish it now?