# Cylinder in a Cone

• Jan 23rd 2007, 03:26 PM
symmetry
Cylinder in a Cone
Inscribed a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. Express the volume V of the cylinder as a function of r.

HINTS GIVEN:

(1) V = pi(r^2)(h).

(2) There are similar triangles in the cone.
• Jan 24th 2007, 05:33 AM
earboth
Quote:

Originally Posted by symmetry
Inscribed a right circular cylinder of height h and radius r in a cone of fixed radius R and fixed height H. Express the volume V of the cylinder as a function of r. ...

Hello,

I've attached a diagram (see attachment)

You have 2 similar right triangles:
- the big one with the legs H and R
- the small on with the legs (H-h) and r.

Set up the proportion:

$\displaystyle \frac{r}{R}=\frac{H-h}{H}$. Solve for r:

$\displaystyle r=\frac{H-h}{H} \cdot R=R-\frac{R}{H} \cdot h$

Plug in this term into the formula of the volume of a cylinder: $\displaystyle V=\pi \cdot r^2 \cdot h$

$\displaystyle V(h)=\pi \cdot \left(R-\frac{R}{H} \cdot h \right)^2 \cdot h$

EB
• Jan 24th 2007, 12:59 PM
symmetry
ok
Absolutely amazing reply and great math notes for my state exam.

Thanks!