Two cylindrical jars have their diameters in the ratio 3 : 1 but heights in…

Concept

For a right circular cylinder, the volume is the area of the circular base multiplied by the height:

V = πr2h

The radius appears squared while the height appears only to the first power. So scaling the radius by a factor k multiplies the volume by k2, whereas scaling the height by a factor m multiplies the volume only by m. A radius is exactly half of its diameter, so two cylinders whose diameters are in a given ratio have their radii in that very same ratio.

Applying it to these two jars

1. The diameters are in the ratio 3 : 1, so the radii are in the ratio 3 : 1 as well. Write r1 = 3r and r2 = r.

2. The heights are in the ratio 1 : 3. Write h1 = h and h2 = 3h.

3. First jar: V1 = π(3r)2(h) = 9πr2h.

4. Second jar: V2 = π(r)2(3h) = 3πr2h.

5. Divide one by the other: V1 : V2 = 9πr2h : 3πr2h = 9 : 3 = 3 : 1.

Cross-check and contrast

Put in concrete numbers: with r1 = 3 and h1 = 1 the first jar gives V1 = 9π, and with r2 = 1 and h2 = 3 the second jar gives V2 = 3π, so V1 : V2 = 9 : 3 = 3 : 1. The wider jar wins because tripling the radius multiplies the volume nine-fold, while tripling the height only triples it.

• Treating the volume as proportional to radius × height instead of (radius)2 × height gives (3 × 1) : (1 × 3) = 1 : 1, because the squaring of the radius has been dropped.

• Comparing the jars by height alone gives 1 : 3, because the difference in cross-section has been left out of the comparison.

• No combination of πr2h with the ratios 3 : 1 and 1 : 3 produces 1 : 4.

Hence the ratio of the volumes is 3 : 1.