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

पर्याय

• 1 : 3

• 1 : 2

• 2 : 1

•  3 : 1

Answer 1

Since,

${\displaystyle \Delta AOV\text{and}LO\prime V}$are similar triangles,

i.e., In ${\displaystyle \Delta AOV\text{and}LO\prime V}$

$$\frac{OA}{O^{\prime }L}=\frac{OV}{O^{\prime }V}$$

$$\Rightarrow \frac{r_1}{r_2}=\frac{h_1}{h_1-h_2}$$

$$\Rightarrow (h_1-h_2)r_1=h_1{r}_2$$

$$\Rightarrow r_1{h}_1-r_1{h}_2=h_1{r}_2$$

$$\Rightarrow r_1{h}_1-h_1{r}_2=r_1{h}_2$$

$$\Rightarrow h_1(r_1-r_2)=r_1{h}_2$$

$$\Rightarrow \frac{(r_1-r_2)}{r_1}=\frac{h_2}{h_1}$$

$$\Rightarrow \frac{(r_1-r_2)}{r_1}=\frac{1}{2}$$

$$\Rightarrow 1-\frac{r_2}{r_1}=\frac{1}{2}$$

$$\Rightarrow \frac{r_2}{r_1}=1-\frac{1}{2}=\frac{1}{2}$$

Thus,  $$r_2:r_1=1:2$$