In the given figure, the area of the equilateral triangle is

Question

Q4_Quant.png In the given figure, the area of the equilateral triangle is 48. If the other three figures are squares, what is the perimeter, approximately, of the nine-sided shape they form? A) 8\sqrt{2} B) 24\sqrt{3} C) 72\sqrt{2} D) 144\sqrt{2} E) 384

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=21070 
In the given figure, the area of the equilateral triangle is 48. If the other three figures are squares, what is the perimeter, approximately, of the nine-sided shape they form?

A)  8\sqrt{2} 
B)  24\sqrt{3} 
C)  72\sqrt{2} 
D)  144\sqrt{2} 
E)  384

The area of equilateral triangle is  side^2*\frac{\sqrt{3}}{4} .

So, we are given that  side^2*\frac{\sqrt{3}}{4}=48  -->  side^2=64\sqrt{3}  -->  side=8\sqrt {3} .

The perimeter =  9*8\sqrt {3}=72\sqrt {3}  -->  \sqrt {3}\approx{\sqrt{2}} .

Answer: C.

madn800 · Answer

Buneul, how did you estimate  fourth\sqrt{3} = \sqrt{2} ?

Bunuel · Answer

side^2=64\sqrt{3}  -->  side^2=8^2*\sqrt{3}  -->  side=\sqrt{8^2*\sqrt{3}} -->  side=8\sqrt {3} .

Hope it's clear.

gmatprav · Answer

Hi Bunuel the question is really how can we understand the value of  3^{1/4} = \sqrt{2}  ??

I was stuck at this point as well. How does  9*8*(3)^{1/4}  convert to  9*8*\sqrt{2}  It may be something very small but I am not able to wrap my head around it.

Bunuel · Answer

The trick here is that any positive integer root from a number more than 1 will be more than 1. For example:  \sqrt {2}>1 .

So, we know that  1<\sqrt {3}<2 . We also know that  \sqrt{2}\approx {1.4} . So,  1<(\sqrt {3}\approx{\sqrt{2}})<2 .

Does this make sense?

gmatprav · Answer

Thanks Bunuel, that makes sense now, coupled with the fact that the question is asking for approximate value not exact value. Thanks!!

Bunuel · Answer

Yes, the fact that we need only the  approximate  value is the key here.

koenh · Answer

This may be really obvious but how is  side^2*\frac{\sqrt{3}}{4}=48  -->  side^2=64\sqrt{3} ? Dividing by fraction is multiplication with the inverse so side^2 would be 48*4/\sqrt{3}?

Bunuel · Answer

side^2*\frac{\sqrt{3}}{4}=48 side^2=\frac{4*48}{\sqrt{3}} Multiply by \frac{\sqrt{3}}{\sqrt{3}} : side^2=\frac{4*48}{\sqrt{3}}*\frac{\sqrt{3}}{\sqrt{3}}=\frac{4*48*\sqrt{3}}{3}=64\sqrt{3} This algebraic manipulation is called rationalization and is performed to eliminate irrational expression in the denominator. Questions involving rationalization to practice: if-x-0-then-106291.html if-n-is-positive-which-of-the-following-is-equal-to-31236.html consider-a-quarter-of-a-circle-of-radius-16-let-r-be-the-131083.html in-the-diagram-not-drawn-to-scale-sector-pq-is-a-quarter-139282.html in-the-diagram-what-is-the-value-of-x-129962.html the-perimeter-of-a-right-isoscles-triangle-is-127049.html which-of-the-following-is-equal-to-98531.html if-x-is-positive-then-1-root-x-1-root-x-163491.html 1-2-sqrt3-64378.html if-a-square-mirror-has-a-20-inch-diagonal-what-is-the-99359.html Hope it helps.

manishtank1988 · Answer

Thanks Bunuel for this list :idea:

Samuelboyle96 · Answer

I think you need to know going into the test that root 2 is approx 1.41 and root 3 is approx 1.73 and root 5 is approx 2.2. So when you root 1.73 which is close to 2 you would get a number also close to root 2.

JeffTargetTestPrep · Answer

Since the area of the equilateral triangle is 48, we can use the following formula to determine the side:

(side^2 x √3)/4 = 48

side^2 = 192/√3

side^2 = 192/√3 x √3/√3

side^2 = 192√3/3

side^2 = 64√3

√side^2 = √(64√3)

side = (8)√(√3)

Since √3 ≈ 1.7, we have √(√3) ≈ √1.7 ≈ 1.3, and we have:

side ≈ (8)(1.3)

Since √2 is about 1.4, we have:

side ≈ 8√2

Thus, the perimeter is approximately 9 x 8√2 = 72√2.

Answer: C

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