A sonographer measures a gallstone as a sphere of diameter 12 millimetres. What is its volume?

Step-by-step solution

1. 1

   Convert diameter to radius first. The sphere formula takes the radius, and using $12$ in place of $6$ would inflate the answer by a factor of $2^3=8$:

   $r=\frac{d}{2}=\frac{12}{2}=6\ \text{mm}$

2. 2

   Write the volume formula.

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

3. 3

   Cube the radius and substitute.

   $6^{3}=216,\qquad V=\frac{4}{3}\pi(216)=288\pi\ \text{mm}^3$

   Keeping $288\pi$ exact until the end avoids compounding rounding error.

4. 4

   Convert to a decimal. With the true value of $\pi$,

   $V=288\pi\approx 904.779\ \text{mm}^3$

   $\boxed{288\pi\approx 904.8\ \text{mm}^3}$

5. 5

   Reconcile with the printed answer key. Many textbooks use the approximation $\pi\approx 3.14$, which gives $288\times 3.14=904.32\ \text{mm}^3$ — that is why an option of $904.32$ appears rather than $904.78$. Both come from the same $288\pi$; the difference is purely the value substituted for $\pi$, so quote $288\pi$ if an exact answer is allowed.

6. 6

   Sanity-check the unit conversion. Since $1\ \text{cm}=10\ \text{mm}$, one cubic centimetre is $1000\ \text{mm}^3$, so the stone is about $0.9\ \text{cm}^3$ — which rules out any option quoted as several cubic centimetres.