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.
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.
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