Question

A right circular cone of height h and radius R is cut by a plane parallel to the base at a distance h/4 from the base. The ratio of the curved surface areas of the resulting cone to the frustum is:

• A. 7 : 16
• B. 16 : 9
• C. 16 : 7
• D. 9 : 7

Answer 1

Answer:

D. 9 : 7

A plane parallel to the base cuts the cone at a height ( \frac{h}{4} ) from the base.

So:

• Height of small top cone $(= h - \frac{h}{4} = \frac{3h}{4})$

• It is similar to the original cone, so the linear scale factor is $( \frac{3}{4} )$

  Curved surface area (CSA)

For similar cones, CSA scales as the square of the linear scale factor.

So,
$\text{CSA}_{\text{small cone}} \propto \left(\frac{3}{4}\right)^2 = \frac{9}{16}$

Thus,

• CSA of small cone = $( \frac{9}{16} )$of CSA of original cone

• CSA of frustum = remaining part = $( 1 - \frac{9}{16} = \frac{7}{16} )$

Required ratio:

• $\text{CSA of cone : CSA of frustum} = \frac{9}{16} : \frac{7}{16} = 9 : 7$