Question

The height and radius of the cone of which the frustum is a part are h1 units and r1 units respectively. Height of the frustum is h2 units and radius of the smaller base is r2  units. If h2  : h1  = 1 : 2 then r1  : r2  is
(1) 1 : 3
(2) 1 : 2
(3) 2 : 1
(4) 3 : 1
​

Answer 1

$r_1:r_2$ = 2:1

Step-by-step explanation:

Radius of bottom of the frustum, OA = $r_1$  

Radius of top of the frustum, O’L = $r_2$ 

Height of a cone , OV = $h_1$

Height of a Frustum, O’O = $h_2$

Height of a smaller cone = O'V = $h_1 - h_2$

In ∆VO’L & ∆VOA,

∠∆VO’L = ∠VOA  (each 90°)

$h_1 - h_2$

∠VLO’ = ∠VAO  (corresponding angles)

∆VO’L ~  ∆VOA  [By AA Similarity]

$\frac{O'L}{OA} =\frac{ O'V}{OV}$

[Corresponding sides of a similar triangles are proportional]

$r_2/r_1 = (h_1 - h_2) /h_1$

$r_2/r_1 = h_1/h_1 - h_2 /h_1$

$r_2/r_1 = 1- h_2 /h_1$

$h_2 /h_1 = 1 - r_2/r_1$

$h_2 /h_1= (r_1 - r_2)/r_1$

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

[$h_2 : h_1 = 1 :2$]

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

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

$r_2/r_1 = 1 -\frac{1}{2}$ 

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

$r_2/r_1 = \frac{1}{2}$

$r_2/r_1 = \frac{1}{2}$

$r_2 : r_1 = 1 : 2$  

Ratio of the radius =$r_2 : r_1$= 1 : 2  

Hence, the ratio of the Radius is $r_2 : r_1$ = 1 : 2  

⇒$r_1:r_2$ = 2:1