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/VolumeVolume 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 DII. 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 \(\sqrt{3}\)
Answer D*
Or \(k * \sqrt{3} = 3\)
\(k = \frac{3}{\sqrt{3}}\)
\(k = \frac{3}{\sqrt{3}} * \frac{\sqrt{3}}{\sqrt{3}}\)
\(k=\frac{3\sqrt{3}}{3}=\sqrt{3}\)
_________________
Do I dare
Disturb the universe? (—T. S. Eliot)
Absolutely.
SC Butler (two Sentence Correction questions per day from Nov 7 to Mar 1) is here.