Hello, Mr_Green!

This is a good problem.

You need to know the volume of a cone: .$\displaystyle V\:=\:\frac{\pi}{3}r^2h$

. . and be able to apply it to their questions.

Given: a paper cone for water is 6 cm across at the top and 10 cm high.

(a) If it is filled to half of its height, what percent of it's total volume does it contain?

(b) If it is filled to half of its volume, what is the height of the water? Code:

: 3 :
- *-------+-------*
: \ | /
: \ | r /
: \----+----/
10 \:::|:::/
: \::|h:/
: \:|:/
: \|/
- *

The volume of the cone is: .$\displaystyle V \:=\:\frac{\pi}{3}\cdot3^2\cdot10\:=\:30\pi$ cm³.

From the similar right triangles: .$\displaystyle \frac{r}{h} = \frac{3}{10}\quad\Rightarrow\quad r \,=\,\frac{3}{10}h$

(a) If $\displaystyle h = 5$, then $\displaystyle r = \frac{3}{10}(5) = \frac{3}{2}$

Then the volume of the water is: .$\displaystyle V \:=\:\frac{\pi}{3}\left(\frac{3}{2}\right)^2(5)\:= \:\frac{15\pi}{4}$ cm³.

The percent of volume is: .$\displaystyle \frac{\frac{15\pi}{4}}{30\pi} \;=\;\frac{1}{8}\;=\;\boxed{12.5\%}$

(b) Since $\displaystyle r = \frac{3}{10}h\!:\;\;V \:=\:\frac{\pi}{3}\!\left(\frac{3}{100}h\right)^2\ !\!h\quad\Rightarrow\quad V\:=\:\frac{3\pi}{100}h^3$

If the cone is half full, the volume of water is $\displaystyle 15\pi$ cm³.

We have: .$\displaystyle \frac{3\pi}{100}h^3\:=\:15\pi\quad\Rightarrow\quad h^3\:=\:500\quad\Rightarrow\quad h \,=\,\sqrt[3]{500}$

Therefore: .$\displaystyle h \;=\;5\sqrt[3]{4}\;\approx\;\boxed{7.937\text{ cm}}$