Question

The height and radius of the cone of which the frustum is a part are h₁ and r₁ respectively. If h₂ and r₂ are the heights and radius of the smaller base of the frustum respectively and h₂ : h₁ = 1 : 2, then r₂ : r₁ is equal to

Options

• 1 : 3

• 1 : 2

• 2 : 1

•  3 : 1

Answer 1

Since,

`Delta AOV "and " LO'V`are similar triangles,

i.e., In `Delta AOV "and " LO'V`

\[\frac{OA}{O'L} = \frac{OV}{O'V}\]

\[ \Rightarrow \frac{r_1}{r_2} = \frac{h_1}{h_1 - h_2}\]

\[ \Rightarrow \left( h_1 - h_2 \right) r_1 = h_1 r_2\]

\[\Rightarrow r_1 h_1 - r_1 h_2 = h_1 r_2 \]

\[ \Rightarrow r_1 h_1 - h_1 r_2 = r_1 h_2 \]

\[ \Rightarrow h_1 \left( r_1 - r_2 \right) = r_1 h_2 \]

\[ \Rightarrow \frac{\left( r_1 - r_2 \right)}{r_1} = \frac{h_2}{h_1}\]

\[ \Rightarrow \frac{\left( r_1 - r_2 \right)}{r_1} = \frac{1}{2}\]

\[ \Rightarrow 1 - \frac{r_2}{r_1} = \frac{1}{2}\]

\[ \Rightarrow \frac{r_2}{r_1} = 1 - \frac{1}{2} = \frac{1}{2}\]

Thus,  \[r_2 : r_1 = 1: 2\]