Question

The height of the frustum of a cone is 12cm and if its slant height is 15cm, then the difference of the radii of the two circular ends is

Answer 1

Let radius of smaller circular end = r
radius of larger circular end = R
height of frustum = h = 12cm
slant height of frustum = l = 15cm

we know, the relation between h, l , r and R
e.g., $\bold{l = \sqrt{(R-r)^2 + h^2}}$
$\bold{15 = \sqrt{(R-r)^2 + 12^2}}$
Squaring both sides,
15² = (R - r)² + 12²
⇒225 = (R - r)² + 144
⇒225 - 144 = (R - r)²
⇒ 81 = (R - r)²
Taking square root both sides,
⇒ ±9 = (R - r) , but (R - r) ≠ negative [ as I assumed R is bigger than r ]
∴ ( R - r) = 9
so, difference between radii = 9

Answer 2

Slant height of the frustum : The slant height of frustum of a right circular cone is the length of the line segment which is obtained by joining the endpoints of two parallel Radii ,drawn in the same direction of the two circular bases.

Height of the frustum : The height of a frustum is a perpendicular distance between its two circular bases.

GIVEN :
Height of frustum (h ) = 12cm
Slant height of frustum ( l ) = 15cm

Let ‘r2’ be the radius of smaller circular end & r1 be radius of larger circular end (r1 > r2)

Slant Height of frustum (l) = √(r1- r2)² +h²

15 = √(r1- r2)² + 12²
15² =(r1- r2)² + 12²
[On Squaring both sides]
225 =(r1- r2)² + 144
225 - 144 =(r1- r2)²
81 =(r1- r2)²
√81 = (r1- r2)
[On Taking square root both sides]
(r1- r2) = ±9
[ (r1- r2) ≠ negative [ as r1 > r12]
Therefore, (r1- r2) = 9

Hence, the difference of the radii of the two circular ends is 9 cm.

HOPE THIS WILL HELP YOU...