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The area of an equilateral triangle with side length x is the same as

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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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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?
[Reveal] Spoiler: OA

Last edited by Skywalker18 on 06 Aug 2017, 11:44, edited 1 time in total.
OA updated

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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, 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

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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

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

I'm confused.

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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, 20:14
Formula needed:
Area of equilateral triangle = $$(\frac{√3}{4})* x^2$$ where x is the area of an equilateral triangle.

Total surface area of a cube = $$6*z^2$$ where z is the side of a cube.

We are given that the areas of equilateral triangle = total surface area of cube.

$$(\frac{√3}{4})* x^2 = 6*z^2$$
$$(\frac{√3}{4})* x^2 = 6*z^2$$
$$x^2 = \frac{6*4*√3*z^2}{√3*√3}$$ (Multiplying and dividing the fraction by √3)
$$x^2 = \frac{6*4*√3*z^2}{3} = 8*√3*z^2$$

The answer has to be Option 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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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 $$\sqrt{3}$$)

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

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

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

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

Can anyone help explain?

$$\sqrt{3}$$/4 * x^2 = 6 * z^2
-> x^2 = (24/$$\sqrt{3}$$)* z^2
-> x^2 =(8*$$\sqrt{3}$$)*z^2

So both x and z can be integers only if x = z = 0 .

So, I think there is no such option and questions needs a reframing.. Bunuel for your reference..
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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, 08:38
The OA need to be changed for this one to 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, 13:53
darn wrote:
The OA need to be changed for this one to B

Or to adjust the length of the triangular from x to x/3.

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

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06 Aug 2017, 02:05
sane here answer should be "B".

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Re: The area of an equilateral triangle with side length x is the same as   [#permalink] 06 Aug 2017, 02:05
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