A jeweler has three small solid spheres made of gold, of radius 1 mm, 2 mm, and 3 mm. He decides to melt these down and make just one sphere out of them. What will the radius of this larger sphere be? Round the answer to one decimal place
A) $4.5 \mathrm {~mm}$
B) $3.5 \mathrm {~mm}$
C) $5.0 \mathrm {~mm}$
D) $6.0 \mathrm {~mm}$
E) none of these

Question

Quizplus · Accepted Answer

The volume of a sphere is given by $V = \frac{4}{3}\pi r^3$. The total volume of the three smaller spheres is the sum of their individual volumes. For the spheres with radii 1 mm, 2 mm, and 3 mm, their volumes are $\frac{4}{3}\pi(1)^3$, $\frac{4}{3}\pi(2)^3$, and $\frac{4}{3}\pi(3)^3$ respectively. Summing these gives $\frac{4}{3}\pi(1^3 + 2^3 + 3^3) = \frac{4}{3}\pi(1 + 8 + 27) = \frac{4}{3}\pi(36)$. The volume of the larger sphere will be the same, so setting $\frac{4}{3}\pi R^3 = \frac{4}{3}\pi(36)$ and solving for $R$ gives $R^3 = 36$, so $R = \sqrt[3]{36}$. Calculating this gives $R \approx 3.3$ mm, which is not listed among the options, making the correct choice "none of these."

A Jeweler Has Three Small Solid Spheres Made of Gold 4.5 mm4.5 \mathrm {~mm}4.5 mm

A Jeweler Has Three Small Solid Spheres Made of Gold $4.5 \mathrm {~mm}$