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If the volume of a cube with side of length 6 is equal

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If the volume of a cube with side of length 6 is equal to the volume of a rectangular solid with length 12 and width 9, then the height of the rectangular solid is

(A) 2
(B) 4
(C) 6
(D) 8
(E) 9

The OA is A.

What is the formula that I should use? And what are the equations needed?

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by ErikaPrepScholar » Thu Mar 15, 2018 5:46 am
To solve, we should know a couple volume formulas. To find volume of a cube, V = s^3, where s is the length of one side. To find volume of a rectangular solid, V = lwh, where l is length, w is width, and h is height.

We know that the volume of the cube is equal to the volume of the rectangular solid, so
s^3 = lwh

We also know that s = 6, l = 12, and w = 9, so
6^3 = (12)(9)h

We solve for h:
216 = 108h
h = 2

which gives is the height of the rectangular solid, or answer choice A.
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by Vincen » Thu Mar 15, 2018 6:51 am
Hello Vjesus12.

We know that the side of the cube is 6, and the volume of a cube is $$V_{cube}=L^3\ \ \ \ \ \Rightarrow\ \ \ \ V_{cube}=\ 6^3=216.$$ Now, this volume is equal to the volume of a rectangular solid with length 12 and width 9. The volume of this rectangular solid is $$V_{\text{rectangular}}=L\cdot W\cdot H=12\cdot9\cdot H=108H.$$ Hence we can set the equation: $$108H\ =216\ \Rightarrow\ H=2.$$ Therefore, the correct answer is the option A.

I hope it helps.

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by DavidG@VeritasPrep » Thu Mar 15, 2018 8:18 am
VJesus12 wrote:If the volume of a cube with side of length 6 is equal to the volume of a rectangular solid with length 12 and width 9, then the height of the rectangular solid is

(A) 2
(B) 4
(C) 6
(D) 8
(E) 9

The OA is A.

What is the formula that I should use? And what are the equations needed?
There's no getting around needing the equations that both Vincen and Erika used in their elegant solutions. If you were hoping to avoid doing arithmetic, once you were here: 6*6*6 = 12*9*h, you could proceed like this
6*6*6 = 2*3 * 2*3 * 2*3 = 2*2*2*3*3*3
12*9*h = 2*2*3*3*3*h
Now we have 2*2*2*3*3*3 = 2*2*3*3*3*h. All we're missing on the right side of the equation is a 2. The answer is A
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by Scott@TargetTestPrep » Mon Mar 19, 2018 5:44 am
VJesus12 wrote:If the volume of a cube with side of length 6 is equal to the volume of a rectangular solid with length 12 and width 9, then the height of the rectangular solid is

(A) 2
(B) 4
(C) 6
(D) 8
(E) 9
The volume of the cube with a side length of 6 is 6 x 6 x 6.

If we let the height of the rectangular solid = h, then we have:

6 x 6 x 6 = 12 x 9 x h

216 = 108h

2 = h

Answer: A

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by swerve » Mon Mar 19, 2018 9:53 am
The volume of the cube with a side length of 6 is 6 x 6 x 6.

If we let the height of the rectangular solid = h, then we have:

6 x 6 x 6 = 12 x 9 x h

216 = 108h

2 = h. Option A.

Regards!