Two cones have their heights in the ratio 1:3 and radii in the ratio 3: 1. What is the ratio of their volumes?
The correct answer and explanation is:
To find the ratio of the volumes of two cones, we use the formula for the volume of a cone: V=13πr2hV = \frac{1}{3} \pi r^2 h
where:
- rr is the radius,
- hh is the height, and
- VV is the volume of the cone.
Let the height of the first cone be h1h_1 and the radius be r1r_1. Let the height of the second cone be h2h_2 and the radius be r2r_2. We are given the ratio of heights as: h1h2=13\frac{h_1}{h_2} = \frac{1}{3}
and the ratio of radii as: r1r2=31\frac{r_1}{r_2} = \frac{3}{1}
The volume of the first cone is: V1=13πr12h1V_1 = \frac{1}{3} \pi r_1^2 h_1
The volume of the second cone is: V2=13πr22h2V_2 = \frac{1}{3} \pi r_2^2 h_2
To find the ratio of the volumes, we divide V1V_1 by V2V_2: V1V2=13πr12h113πr22h2\frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2}
The 13π\frac{1}{3} \pi terms cancel out, leaving: V1V2=r12h1r22h2\frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2}
Substituting the given ratios r1r2=3\frac{r_1}{r_2} = 3 and h1h2=13\frac{h_1}{h_2} = \frac{1}{3}, we get: V1V2=(3)2×1(1)2×3=93=3\frac{V_1}{V_2} = \frac{(3)^2 \times 1}{(1)^2 \times 3} = \frac{9}{3} = 3
Therefore, the ratio of the volumes of the two cones is: 3:13:1
Explanation:
The volume of a cone is directly proportional to the square of the radius and the height. When the ratio of the radii is 3:1, the square of the radius ratio becomes 32=93^2 = 9. Since the ratio of the heights is 1:3, the height ratio contributes a factor of 13\frac{1}{3}. Combining these factors, we get a final ratio of 9×13=39 \times \frac{1}{3} = 3, so the volume ratio is 3:1.