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CAT 2020 Slot 1 — QA Question 16

Basics of Mensuration/PrismEasy

A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a height of 18 cm from the base. If the difference in volume of the two pieces is 225 cc, the volume, in cc, of the original cone is

Answer & solution

  • A

    256

  • B

    232

  • C

    264

  • 243

Solution

Easy

The small top cone is similar to the whole cone, so volumes scale as the cube of the height ratio. The cut at height 1818 from the base leaves a top cone of height 27−18=927-18=9. Express both pieces (small cone and frustum) in terms of the small-cone volume, then use the volume difference =225=225.

top cone, h = 9 frustum cut @ 18 from base
1

Similar-cone volume ratio. The top cone has height 99, the whole cone height 2727.

VtopVwhole=(927)3=(13)3=127\begin{aligned} &\frac{V_{\text{top}}}{V_{\text{whole}}} = \left(\frac{9}{27}\right)^3 = \left(\frac13\right)^3 = \frac{1}{27} \end{aligned}
2

Name the pieces. Let the small top cone have volume xx. Then the whole cone is 27x27x and the frustum (lower piece) is the remainder.

Vfrustum=27x−x=26x\begin{aligned} &V_{\text{frustum}} = 27x - x = 26x \end{aligned}
3

Use the difference of 225225. The two pieces are the frustum 26x26x and the top cone xx.

26x−x=225⇒ 25x=225⇒ x=9\begin{aligned} &26x - x = 225\\ &\Rightarrow\ 25x = 225\\ &\Rightarrow\ x = 9 \end{aligned}
4

Original cone volume.

Vwhole=27x=27×9=243\begin{aligned} &V_{\text{whole}} = 27x = 27\times 9 = 243 \end{aligned}
243 cc— option (d)243\ \text{cc}\quad\text{— option (d)}
CAT 2020 Slot 1 QA Q16: A solid right circular cone of height 27 cm is cut into two pieces along a plane parallel to its base at a hei — Solution | TheCATExam