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# The volume of a cube with edge 3 is how many times the volume of a cub

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The volume of a cube with edge 3 is how many times the volume of a cub  [#permalink]

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15 Apr 2018, 10:22
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The volume of a cube with edge 3 is how many times the volume of a cube with edge $$\sqrt{3}$$?

a. $$\frac{1}{3}$$

b. $$1$$

c. $$3$$

d. $$3\sqrt{3}$$

e. $$9$$

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The volume of a cube with edge 3 is how many times the volume of a cub  [#permalink]

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15 Apr 2018, 12:51
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QZ wrote:
The volume of a cube with edge $$3$$ is how many times the volume of a cube with edge $$\sqrt{3}$$?

a. $$\frac{1}{3}$$

b. $$1$$

c. $$3$$

d. $$3\sqrt{3}$$

e. $$9$$

I. Volume/Volume

Volume of a cube = $$s^3$$
Volume of smaller cube = $$(\sqrt{3})^3 = 3\sqrt{3}$$
Volume of larger cube = $$3^3 = 27$$

Volume of $$27$$ is how many times greater than $$3\sqrt{3}$$?

$$\frac{27}{3\sqrt{3}} =$$

$$\frac{27}{3\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}=$$

$$\frac{27\sqrt{3}}{9} = 3\sqrt{3}$$

II. Cube the scale factor

• SCALE FACTOR: If a shape's size increases or decreases, it scales up or scales down.
That means that every length in the shape has been multiplied by a scale factor, $$k$$

The scale factor is a multiplier; any change in length = length * scale factor $$k$$

Scale factors tell you "how many times"
the smaller size was multiplied to obtain the greater size

You need nothing else in this problem except (scale factor)$$^3$$

• To account for change in length, area, or volume:
Length = $$k$$
Area = (length * length) = $$k^2$$
Volume= (length * length * length) = $$k^3$$

• Scale factor here?
Use ONE length's increase to find $$k$$:
$$(k) * (s$$ of small cube) = ($$s$$ of large cube)

Scale factor EQUALS?*
Small cube's side length: $$\sqrt{3}$$
Large cube's side length = 3
$$k * \sqrt{3} = 3$$
$$k=\frac{3}{\sqrt{3}}=\frac{3^1}{3^{\frac{1}{2}}}=3^{(1-\frac{1}{2})}=3^{\frac{1}{2}}$$
$$k=3^{\frac{1}{2}}=\sqrt{3}$$

Volume increase? $$k^3$$
To find out "how many times greater," because it's a volume change --
cube the scale factor
$$(\sqrt{3})^3 = (\sqrt{3} * \sqrt{3} * \sqrt{3}) = 3\sqrt{3}$$

The volume of a cube with edge $$3$$ is $$3\sqrt{3}$$ times the volume of a cube with edge $$\sqrt{3}$$

*Or $$k * \sqrt{3} = 3$$
$$k = \frac{3}{\sqrt{3}}$$
$$k = \frac{3}{\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}$$
$$k=\frac{3\sqrt{3}}{3}=\sqrt{3}$$

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Re: The volume of a cube with edge 3 is how many times the volume of a cub  [#permalink]

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19 Feb 2019, 09:39
I don't think this is a sub-600 question.

We can use the rule "if we have 2 solids with the ratio between 2 sides a and b, the volume of the solids will be in the ratio of a^3:b^3.

So 3^3 : sqrt(3)^3 --> 27 : 3*sqrt(3) --> 9 : sqrt(3)

The question asks what is the multiplier of sqrt(3) that equals 9, i.e. x*sqrt(3) = 9
so x = 3*sqrt(3), D.
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Re: The volume of a cube with edge 3 is how many times the volume of a cub  [#permalink]

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21 Feb 2019, 18:08
AkshdeepS wrote:
The volume of a cube with edge 3 is how many times the volume of a cube with edge $$\sqrt{3}$$?

a. $$\frac{1}{3}$$

b. $$1$$

c. $$3$$

d. $$3\sqrt{3}$$

e. $$9$$

The volume of a cube with edge 3 is 3^3 = 27

The volume of a cube with edge √3 is (√3)^3 = 3√3.

Thus, the volume with edge 3 is 27/(3√3) = 9/√3 = 9√3/3 = 3√3 times the volume of the cube with edge √3.

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Re: The volume of a cube with edge 3 is how many times the volume of a cub   [#permalink] 21 Feb 2019, 18:08
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