Question

If the radius of a sphere is increased to twice its original size, what is the ratio of the new surface area to the original surface area, as well as the proportion of the new volume to the original volume?

• A. 4:1 and 2:1
• B. 4:1 and 8:1
• C. 8:1 and 4:1
• D. 2:1 and 4:1

Answer 1

Answer:

B. 4:1 and 8:1

For a sphere:

• Surface Area (= 4\pi r^2)

• Volume (= \frac{4}{3}\pi r^3)

If the radius is doubled:

$r \rightarrow 2r$

Surface Area

New surface area:

$4\pi(2r)^2 = 4\pi \cdot 4r^2 = 16\pi r^2$

Original surface area:

$4\pi r^2$

Ratio:

$16\pi r^2 : 4\pi r^2 = 4:1$

Volume

New volume:

$\frac{4}{3}\pi(2r)^3$
$= \frac{4}{3}\pi \cdot 8r^3$

Original volume:

$\frac{4}{3}\pi r^3$

Ratio:

$8:1$

Answer:

• New Surface Area : Original Surface Area = $(\boxed{4:1})$

• New Volume : Original Volume = $(\boxed{8:1})$