The digital volume of a cone beam x-ray scan is 2616 cm³. To capture a craniofacial image on the detector, the base of the cone beam on the detector must have a diameter of 20 cm. What should the distance be from the source to the detector (i.e., the height of the cone beam)? Use π=3.14 and round your answer to the nearest whole number.

The correct answer is:

25 cm.

Explanation:

The volume of a cone is given by the formula
[tex]V=\frac{\pi r^2h}{3}[/tex]

The diameter is 20 cm, so the radius is half of that:
20/2=10 cm.

Our volume is 2616 and we will use 3.14 for pi:
[tex]2616=\frac{3.14\times 10^2 \times h}{3}[/tex]

Simplifying the right hand side, we have
[tex]2616=\frac{3.14 \times 100 \times h}{3} \\ \\2616=\frac{314h}{3}[/tex]

Multiply both sides by 3:
[tex]2616\times 3=\frac{314h}{3}\times 3 \\ \\7848=314h[/tex]

Divide both sides by 314:
[tex]\frac{7848}{314}=\frac{314h}{314} \\ \\24.99=h \\ \\25\approx h[/tex]