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Saturday, June 13, 2015

UPSC Maths Problems Challenge : A Problem A Day! (Day 2)

First of all,Thank You to all for the prompt responses and attention I have been receiving on UPSC Maths Problem Challenge Posts!The second day of the challenge consists of a direct problem from mensuration.Read on to find out what.

Today's Problem (14/06/2015)

Two cubes of sides 6 cm each are kept side by side to form a ractangular parallelepiped. the area of whole surface of the ractangular parrallelepiped is?

Question Credit :

The second problem is a direct question from mensuration and though it may appear to be a 4 year old kid's problem,it is not.Either you have to sum up two cubes being placed aside to form a cuboid or you have to subtract the surface area dismissed due to joining the
three-dimensional objects.

When two cubes of equal given measurements are kept aligned together,it forms a cuboid with length = 12 cm,breadth = 6 cm and height = 6 cm.So,the area of the parallelpiped is 2(lb + bh + lh) = 2 (12.6 + 6.12 + 6.6) cm^2 = 2 (72 + 72 + 26) cm^2 = 2 (180) cm^2 = 360 cm^2.

Note that,l = length,b = breadth and h = height of the cuboid and 2 (lb + bh + lh) is the formula of getting the total surface area of a cuboid.

A little diagram to help you understand the concept of the sum.Please excuse the angle in which it has been shown.
(To be honest,I was trying to make these diagrams on an online isometric sheet and they would not allow the y and z axes to be turned and saved but only to be saved the way you have drawn them.You can try drawing and analysing on isometric sheets here.)

Alternate Way of Solving This Problem

While joining the two cubes side by side forms a cuboid,it gets rid of two surfaces,each from one cube and each measuring (6^2)cm^2 = 36 cm^2.
So,the total surface area = 2(6l^2 - l^2) cm^2 = 2(5l^2) cm^2 = 10.l^2 cm^2 = (10 * 36)cm^2
= 360 cm^2

Note that,l = length of each side of a cube.
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