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# [Smart Math] Geometry Problem 10

Here’s and example of a SMART MATH problem for GEOMETRY.

### Problem

The area of the four walls of a room is 1080 sq.ft. If the height and length of the room are in the ratio of 2 : 5 and the height and breadth are in the ratio 4 : 5. What is the area of the roof?

1. 240 sq.ft.
2. 450 sq.ft.
3. 540 sq.ft.
4. 720 sq.ft.
5. Cannot be determined

### The Usual Method

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Let the height and breadth of the room be 2x and 5x respectively.

Similarly let the height and breadth of the room be 4y and 5y respectively.

Area of the walls = length x height x 2 + breadth x height x 2 = 1080

$2x\times 5x\times 2+4y\times 5y\times 2=1080$

$\therefore 20x^{2}+40y^{2}=1080$

$\therefore x^{2}+2y^{2}=54$                                   ……Eq. 1

Also note that height of the room is constant, so:

$2x=4y$

$\therefore x=2y$

Substituting this in Eq. 1, we get:

$\left( 2y \right)^{2}+2y^{2}=54$

$\therefore 6y^{2}=54$

$\therefore y=3$

$\therefore x=2\times 3=6$

Hence, length of room = $5x=5\times 6=30$ft

And breadth of room = $5y=5\times 3=15$ft

Hence area of roof = 30 x 15 = 450 sq.ft.

(Ans: 2)

Estimated Time to arrive at the answer = 90 seconds.

### Using Technique

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Ratio of height to length = 2 : 5 = 4 : 10

Ratio of height to breadth = 4 : 5

Thus ratio of length to breadth = 10 : 5

Hence, area of ceiling = $10x\times 5x=50x^{2}$

Hence, the area of the ceiling should be a multiple of 50. The only option satisfying this condition is option ‘2’.

(Ans: 2)

Estimated Time to arrive at the answer = 10 seconds.
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