The diameter of a ball is 6 inches. What is the volume of the ball? Choose from 12.76, 37.68, 113.04 or 676.91 cubic inches.

Step-by-step solution

1. 1

   Convert the diameter to a radius before opening the formula. The volume formula is written in terms of $r$, and the problem hands you $d$:

   $r=\frac{d}{2}=\frac{6}{2}=3\ \text{in}.$

   Skipping this step is the trap: using $6$ in place of $3$ multiplies the answer by $2^3=8$, which is exactly how the distractor $904$-sized values are generated.

2. 2

   State the formula for the volume of a sphere.

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

3. 3

   Cube the radius, then multiply.

   $r^{3}=3^{3}=27,\qquad V=\frac{4}{3}\pi\cdot 27=\frac{108}{3}\pi=36\pi\ \text{in}^3.$

   The $3$ in the denominator divides into $27$ cleanly, so no decimals are needed yet.

4. 4

   Evaluate numerically. With $\pi\approx 3.14159$,

   $V=36\pi\approx 113.10\ \text{in}^3,$

   and with the classroom value $\pi=3.14$ the same product is $36\times 3.14=113.04\ \text{in}^3$. That is choice C — the small gap between $113.04$ and $113.10$ is purely the rounding of $\pi$, not an error.

5. 5

   Rule out the other options. $37.68=12\pi$ is $4\pi r^2/\ldots$ — in fact it is $\tfrac43\pi r^3$ computed with $r^3$ replaced by $r$; $12.76$ is far too small for a ball three inches in radius; and $676.91$ is roughly $\tfrac43\pi\cdot 6^3/\ldots$, the value you get by forgetting to halve the diameter. Only C survives.