# Thread: a cylinder is inscribed in a cone

1. ## a cylinder is inscribed in a cone

A cylinder is inscribed in a cone. The cylinder has an altitude of 15 cm and a radius of 3 cm. Express H as a function of R.

Thanks!

2. Hello, katinthehat108!

I think I know what you meant . . .

A cylinder is inscribed in a cone.
The cylinder has an altitude of 15 cm and a radius of 3 cm.
Express H (height of the cone) as a function of R (radius of the cone).
Code:
    -           *           -
:          /|\          :
:         / |:\        H-15
:        /  |::\        :
:       *---+---*       -
H      /|   | 3 |\      :
:     / |   |   |:\    15
:    /  |   | 15|::\    :
:   /   |   |   |:::\   :
:  /    |   |   |::::\  :
- *-----*---+---*-----* -
: 3 : R-3 :
: -- R -- :

The two shaded triangles are similar.

Hence, we have: . $\frac{H-15}{3} \:=\:\frac{15}{R-3} \quad\Rightarrow\quad (H-15)(R-3) \:=\:45$

. . . . $HR - 3H - 15R + 45 \:=\:45 \quad\Rightarrow\quad HR - 3H \:=\:15R$

. . . . $H(R-3) \:=\:15R \quad\Rightarrow\quad\boxed{ H \:=\:\frac{15R}{R-3}}$