  Question

A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is __________.

Solution

It is given that a cone, a hemisphere and a cylinder stand on equal bases and have the same height. Let the radius of cone, hemisphere and cylinder be r units. Radius of the cone = Radius of hemisphere = Radius of cylinder = r Also, Height of the cone = Height of the cylinder = Height of the hemisphere We know that, the height of a hemisphere is same as its radius. ∴ Height of the hemisphere = r ⇒ Height of the cone = Height of the cylinder = Height of the hemisphere = r Now, Volume of the cone = $\frac{1}{3}\mathrm{\pi }$ × (Radius)2 × Height = $\frac{1}{3}\mathrm{\pi }$ × r2 × r = $\frac{1}{3}\mathrm{\pi }$r3 Volume of the hemisphere = $\frac{2}{3}\mathrm{\pi }$ × (Radius)3 = $\frac{2}{3}\mathrm{\pi }$r3 Volume of the cylinder = $\mathrm{\pi }$ × (Radius)2 × Height = $\mathrm{\pi }$ × r2 × r = $\mathrm{\pi }$r3 ∴ Volume of the cone : Volume of the hemisphere : Volume of the cylinder = $\frac{1}{3}\mathrm{\pi }$r3 : $\frac{2}{3}\mathrm{\pi }$r3 : $\mathrm{\pi }$r3 = $\frac{1}{3}$ : $\frac{2}{3}$ : 1 = 1 : 2 : 3 Thus, the ratio of their volumes is 1 : 2 : 3. A cone, a hemisphere and a cylinder stand on equal bases and have the same height. The ratio of their volumes is ____1 : 2 : 3____.MathematicsRD Sharma (2019)All

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