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The area of an equilateral triangle with side length x is the same as [#permalink]

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04 Aug 2017, 08:58

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43% (01:11) correct
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The area of an equilateral triangle with side length x is the same as the surface area of a cube with edge length z. If both x and z are integers, then what is x^2 in terms of z^2 ?

A. Z^2 (6√3) B. Z^2 (8√3) C. 72 Z^2 D. Z^2 (72√3) E. 192 Z^2

Re: The area of an equilateral triangle with side length x is the same as [#permalink]

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04 Aug 2017, 09:16

wbricker3 wrote:

The area of an equilateral triangle with side length x is the same as the surface area of a cube with edge length z. If both x and z are integers, then what is x^2 in terms of z^2 ?

A. Z^2 (6√3) B. Z^2 (8√3) C. 72 Z^2 D. Z^2 (72√3) E. 192 Z^2

Can anyone help explain?

Area of equilateral triangle = (√3/4 )* x^2

Total surface area of a cube = 6*z^2

Since the areas of equilateral triangle and total surface area of cube are equal =>(√3/4 )* x^2 = 6*z^2 => x^2 = (24/√3) *z^2 = 8√3 *z^2

Shouldn't B be the answer?
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Re: The area of an equilateral triangle with side length x is the same as [#permalink]

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04 Aug 2017, 17:17

Skywalker18 wrote:

wbricker3 wrote:

The area of an equilateral triangle with side length x is the same as the surface area of a cube with edge length z. If both x and z are integers, then what is x^2 in terms of z^2 ?

A. Z^2 (6√3) B. Z^2 (8√3) C. 72 Z^2 D. Z^2 (72√3) E. 192 Z^2

Can anyone help explain?

Area of equilateral triangle = (√3/4 )* x^2

Total surface area of a cube = 6*z^2

Since the areas of equilateral triangle and total surface area of cube are equal =>(√3/4 )* x^2 = 6*z^2 => x^2 = (24/√3) *z^2 = 8√3 *z^2

Shouldn't B be the answer?

That's exactly that I got. There isn't a scale factor problem here (we have two one-dimensional lengths that factor into a formula for areas, areas that are equal).

Answer D seems to imply . . . Volume of cube? How do you get from here:

\(\frac{24}{\sqrt{3}}\)

to here:

\(\frac{216}{\sqrt{3}}\)????

If you rationalize the denominator, the answer becomes what choice D lists.

Prime factorization of 216 is \(2^3*3^3\). 216/24 = 9. So we missed a scale factor of \(3^2\)?

This OA seems really odd in light of a similar problem with an answer I calculated exactly as I did here. (And, unlike here, I think in that problem there might be a scale factor issue! It involves a length of x meters and an area of x square meters, and it's very similar.) See https://gmatclub.com/forum/if-the-area- ... l#p1901581

Re: The area of an equilateral triangle with side length x is the same as [#permalink]

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04 Aug 2017, 21:57

wbricker3 wrote:

The area of an equilateral triangle with side length x is the same as the surface area of a cube with edge length z. If both x and z are integers, then what is x^2 in terms of z^2 ?

A. Z^2 (6√3) B. Z^2 (8√3) C. 72 Z^2 D. Z^2 (72√3) E. 192 Z^2

Can anyone help explain?

Formula for Area of equilateral triangle with side \(x = \frac{x^2 \sqrt{3}}{4}\)

Formula for Surface area of cube with edge length \(z = 6z^2\)

Given Area of equilateral triangle and Surface area of cube are equal. Therefore,

\(\frac{x^2 \sqrt{3}}{4} = 6z^2\)

\(x^2 = \frac{(z^2)(6 * 4) }{\sqrt{3}}\)

\(x^2 = \frac{(z^2)(6 * 4)(\sqrt{3})}{\sqrt{3}*\sqrt{3}}\) --------- (Multiplying numerator and denominator by \(\sqrt{3}\))

\(x^2 = \frac{(z^2)(6 * 4)(\sqrt{3})}{3}\)

\(x^2 = (z^2)(2 * 4) (\sqrt{3})\)

\(x^2 = (z^2)(8 \sqrt{3})\)

Answer (B)...

I am also getting answer as B...

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Re: The area of an equilateral triangle with side length x is the same as [#permalink]

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05 Aug 2017, 00:28

wbricker3 wrote:

The area of an equilateral triangle with side length x is the same as the surface area of a cube with edge length z. If both x and z are integers, then what is x^2 in terms of z^2 ?

A. Z^2 (6√3) B. Z^2 (8√3) C. 72 Z^2 D. Z^2 (72√3) E. 192 Z^2

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