Question

Two spheres have a volume of 121 cm^3 and 3241 cm^3. What is the ratio of the radius of the small sphere to the radius of the large sphere?

A. 1:5
B. 2:3
C. 3:4
D. 4:1

Answer 1

3:4 is the ratio of the radius of the small sphere to the radius of the large sphere.Thus option c is the correct answer.

Step-by-step explanation:

The ratio of the radii of two spheres is determined by the cube root of the ratio of their volumes. The formula for the volume of a sphere is
`\(V = (4)/(3)πr^3\)`, where \(r\) is the radius. Given the volumes
`\(V_1 = 121 \, \text{cm}^3\)` and
`\(V_2 = 3241 \, \text{cm}^3\)`, the ratio of their volumes is \(V_1 : V_2 = \frac{121}{3241}\).

Now, to find the ratio of their radii, we take the cube root of this volume ratio:
`\(\left((r_1)/(r_2)\right)^3 = (V_1)/(V_2)\)`. Substituting the given volumes, we have
`\(\left((r_1)/(r_2)\right)^3 = (121)/(3241)\)`. Taking the cube root of both sides gives
`\((r_1)/(r_2) = \sqrt[3]{(121)/(3241)}\)`.

Finally, simplifying this expression provides the ratio of the radii:
`\((r_1)/(r_2) = (11)/(31)\)`. To express this ratio in the given answer choices (3:4), we multiply both the numerator and denominator by 3, yielding the equivalent ratio \(\frac{33}{93}\), which simplifies to 3/4. 3:4 is the ratio of the radius of the small sphere to the radius of the large sphere.Thus option c is the correct answer.