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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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New post 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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New post 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 \(3\sqrt{3}\)?

\(\frac{27}{3\sqrt{3}} =\)

\(\frac{27}{3\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}=\)

\(\frac{27\sqrt{3}}{9} = 3\sqrt{3}\)


ANSWER D

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 \(\sqrt{3}\)

Answer D

*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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New post 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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New post 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.

Answer: D
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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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