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CAT 2019 Slot 2 — QA Question 24

Basics of Mensuration/PrismEasy

A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of solid right circular cylinders of each type of metal. These cylinders have the same volume and each of these has radius 3 cm. If the total number of cylinders is to be kept at a minimum, then the total surface area of all these cylinders, in sq cm, is

Answer & solution

  • A

    1044(4 + π)

  • B

    8464π

  • C

    928π

  • 1026(1 + π)

Solution

Easy

Equal-volume cylinders mean equal volumes carved from each metal. To keep the count minimum, make each cylinder as large as possible: the common volume must be the HCF of the three metal volumes. Then count cylinders per metal and add up the lateral + top/bottom areas.

1

Maximum common volume = HCF. Each cylinder has the same volume VV, and VV must divide all three metal volumes. Largest such VV is the HCF.

405=34⋅5,783=33⋅29,351=33⋅13⇒ V=HCF⁡(405,783,351)=33=27 cc\begin{aligned} &405 = 3^4\cdot 5,\quad 783 = 3^3\cdot 29,\quad 351 = 3^3\cdot 13\\ &\Rightarrow\ V=\operatorname{HCF}(405,783,351)=3^3=27\ \text{cc} \end{aligned}
2

Use the volume to fix πh\pi h. With radius r=3r=3:

πr2h=27(volume of one cylinder)⇒ π⋅9⋅h=27⇒ πh=3\begin{aligned} &\pi r^2 h = 27 \quad\text{(volume of one cylinder)}\\ &\Rightarrow\ \pi\cdot 9\cdot h = 27\\ &\Rightarrow\ \pi h = 3 \end{aligned}
3

Count the cylinders. Divide each metal volume by 2727.

Iron=40527=15,Aluminium=78327=29,Copper=35127=13⇒ N=15+29+13=57\begin{aligned} &\text{Iron}=\tfrac{405}{27}=15,\quad \text{Aluminium}=\tfrac{783}{27}=29,\quad \text{Copper}=\tfrac{351}{27}=13\\ &\Rightarrow\ N = 15+29+13 = 57 \end{aligned}
4

Total surface area. Each cylinder has surface area 2πr2+2πrh2\pi r^2+2\pi r h. Multiply by N=57N=57 and substitute r=3r=3, πh=3\pi h=3.

TSA=57(2πr2+2πrh)⇒ =57(2π⋅9+2⋅3⋅πh)(from step 2)⇒ =57(18π+6⋅3)⇒ =57(18π+18)=57⋅18(π+1)\begin{aligned} &\text{TSA}=57\left(2\pi r^2+2\pi r h\right)\\ &\Rightarrow\ =57\left(2\pi\cdot 9 + 2\cdot 3\cdot \pi h\right) \quad\text{(from step 2)}\\ &\Rightarrow\ =57\left(18\pi + 6\cdot 3\right)\\ &\Rightarrow\ =57\left(18\pi + 18\right)=57\cdot 18(\pi+1) \end{aligned}
TSA=1026 (1+π) sq cm\text{TSA}=1026\,(1+\pi)\ \text{sq cm}
CAT 2019 Slot 2 QA Q24: A man makes complete use of 405 cc of iron, 783 cc of aluminium, and 351 cc of copper to make a number of soli — Solution | TheCATExam