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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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Joined: 13 Apr 2013
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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. $$9$$
[Reveal] Spoiler: OA

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

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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The volume of a cube with edge 3 is how many times the volume of a cub   [#permalink] 15 Apr 2018, 12:51
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