The heights of a cone and a cylinder are equal. The radii of their bases are in the ratio 2:1. The ratio of their volumes is:
4:3
Let \(h\) be the height of both the cone and the cylinder.
Let \(r_c\) be the radius of the base of the cone and \(r_{cy}\) be the radius of the base of the cylinder.
We are given that the ratio of their radii is 2:1, so we can write this as:
\(\frac{r_c}{r_{cy}} = \frac{2}{1}\)
This means \(r_c = 2r_{cy}\).
The volume of a cone is given by the formula:
\(V_{cone} = \frac{1}{3}\pi r_c^2 h\)
The volume of a cylinder is given by the formula:
\(V_{cylinder} = \pi r_{cy}^2 h\)
Now, let's find the ratio of their volumes:
\(\frac{V_{cone}}{V_{cylinder}} = \frac{\frac{1}{3}\pi r_c^2 h}{\pi r_{cy}^2 h}\)
Since \(h\) is the same for both, it cancels out:
\(\frac{V_{cone}}{V_{cylinder}} = \frac{\frac{1}{3}r_c^2}{r_{cy}^2} = \frac{1}{3} \left(\frac{r_c}{r_{cy}}\right)^2\)
Substituting \(r_c = 2r_{cy}\):
\(\frac{V_{cone}}{V_{cylinder}} = \frac{1}{3} \left(\frac{2r_{cy}}{r_{cy}}\right)^2 = \frac{1}{3}(2)^2 = \frac{4}{3}\)
Therefore, the ratio of their volumes is 4:3.
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