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Question 24 : 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

  1. 1044(4 + π)
  2. 8464π
  3. 928π
  4. 1026(1 + π)

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Explanatory Answer

Fisrtly, the three materials are not reduced, mixed and then moulded.
Instead a cylinder is formed with one material and one material only.

The base area of the cylinders to be casted is the same(because the base radius is the same)...
Which is \\pi 3^{2} = 9 \pi).

Let's us imagine that each sphere is casted into a single cylinder,
In this case we will have 3 cylinders, each made up of a different material and having different heights.

So, the heights of these cylinders will be \\frac{405}{9 \pi}, \frac{783}{9 \pi}, \frac{351}{9 \pi}=\frac{45}{\pi}, \frac{87}{\pi}, \frac{39}{\pi}).

But we need to have cylinders with equal volume.
Since the base area is the same, this translates to the condition that the cylinders should have the same height. And they should be as high as possible, because the question also wants us to have as less number of cylinders as possible...

So, we need the height of each cylinder to be HCF of (\\frac{45}{\pi}, \frac{87}{\pi}, \frac{39}{\pi})) = \\frac{3}{\pi}).
this implies there will be a total of of 15+29+13 = 57 cylinders.

The total surface area of one such Cylinder is \2 \pi r^{2}+2 \pi r h=2 \pi 3^{2}+2 \pi 3 \frac{3}{\pi}=18 \pi+18=18(\pi+1)).
The TSA of all the 57 cylinders is 57*\18(\pi+1)) = \1026(\pi+1)).


The question is "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"

Hence, the answer is 1026(1 + π)

Choice D is the correct answer.

 

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