Scaling in Space Math Example 2

Follow the full solution, then compare it with the other examples linked below.

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A sphere has radius 2 cm and volume

V = \frac{4}{3}\pi r^3

. If the radius is tripled, how many times larger is the new volume?

1
Step 1: Scale factor $k = 3$ .
2
Step 2: Volume scales by $k^3 = 3^3 = 27$ .
3
Step 3: Original volume: $V_1 = \frac{4}{3}\pi(2)^3 = \frac{32\pi}{3}$ cm³.
4
Step 4: New volume: $V_2 = \frac{4}{3}\pi(6)^3 = \frac{4}{3}\pi(216) = 288\pi$ cm³.
5
Step 5: Ratio: $288\pi \div \frac{32\pi}{3} = 288 \times \frac{3}{32} = 27$ .

Answer

The new volume is

27

times larger.

Volume involves three length dimensions, so scaling lengths by

k

scales volume by

k^3

. Tripling the radius makes the sphere

3^3 = 27

times more voluminous. This cube law has major implications in biology — large animals have proportionally smaller surface-area-to-volume ratios.

About Scaling in Space

How length, area, and volume measurements change when a figure is uniformly enlarged or shrunk by a scale factor.

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