TheCATExam
Sign in

CAT 2017 Slot 1 — QA Question 18

Basics of Mensuration/PrismEasy

A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8 : 27 : 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to:

Answer & solution

Correct answer: 50

  • A

    10

  • 50

  • C

    60

  • D

    20

Solution

Easy

Volume is conserved, so the original cube's volume equals the sum of the five small volumes. Express everything in "parts": cube-roots of volumes give sides, squares of sides give surface areas. Then compare totals.

1

Find the original cube's volume share. The five volumes sum to 1+1+8+27+27=641+1+8+27+27=64, so the original cube is 6464 on the same scale.

volumes = 1:1:8:27:27:64(last = original)\begin{aligned} &\text{volumes}\ =\ 1:1:8:27:27:64 \quad\text{(last = original)} \end{aligned}
2

Sides, then surface-area parts. Side ∝V1/3\propto V^{1/3}, surface area ∝side2\propto \text{side}^2:

sides = 1:1:2:3:3:4(cube roots)areas = 1:1:4:9:9:16(squares)\begin{aligned} &\text{sides}\ =\ 1:1:2:3:3:4 \quad\text{(cube roots)}\\ &\text{areas}\ =\ 1:1:4:9:9:16 \quad\text{(squares)} \end{aligned}
3

Compare the totals. Five small cubes vs. the original:

five small=1+1+4+9+9=24 partsoriginal=16 parts\begin{aligned} &\text{five small}=1+1+4+9+9=24\ \text{parts}\\ &\text{original}=16\ \text{parts} \end{aligned}
4

Percentage excess. Compare the surplus to the original:

24−1616×100=816×100=50%\begin{aligned} &\frac{24-16}{16}\times 100=\frac{8}{16}\times 100=50\% \end{aligned}
Excess≈50%\text{Excess}\approx 50\%

Related Basics of Mensuration/Prism questions

See all Mensuration questions →
CAT 2017 Slot 1 QA Q18: A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8 : 27 : 27. T — Solution | TheCATExam