# Thread: Cylinder in a Cone

1. ## 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.

2. 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:

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

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

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

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

EB

3. ## ok

Absolutely amazing reply and great math notes for my state exam.

Thanks!

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# cone of cylinder

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