Question

A cone, a hemisphere and a cylinder stands on equal bases and have the same height. What is the ratio of their volumes :

• A. 1 : 2 : 3
• B. 2 : 3 : 4
• C. 1 : 3 : 4
• D. none of these

Hint:

Given: Cone, Hemisphere, Cylinder have equal bases (radius \(R\)) and same height (\(H\)). Crucial point: For a hemisphere, its height (when on its flat base) IS its radius. So, \(H = R\). This means for all three shapes, their radius is \(R\) and their height is also \(R\).
\(V_{\text{cone}} = \frac{1}{3}\pi R^2 H = \frac{1}{3}\pi R^2 (R) = \frac{1}{3}\pi R^3\)
\(V_{\text{hemisphere}} = \frac{2}{3}\pi R^3\)
\(V_{\text{cylinder}} = \pi R^2 H = \pi R^2 (R) = \pi R^3\) Ratio: \( \frac{1}{3}\pi R^3 : \frac{2}{3}\pi R^3 : \pi R^3 \) Divide by \(\pi R^3\): \( \frac{1}{3} : \frac{2}{3} : 1 \) Multiply by 3: \( 1 : 2 : 3 \).

Answer 1

The Correct Option is A

Concept: This problem requires knowing the volume formulas for a cone, a hemisphere, and a cylinder, and applying the given conditions about equal bases and same height. Step 1: Define dimensions based on conditions "Equal bases" means they all have the same radius, let it be \(r\). "Same height" means they all have the same height, let it be \(h\). For a hemisphere standing on its circular base, its height is equal to its radius. So, if the hemisphere has the same height \(h\) as the cylinder and cone, then for the hemisphere, its radius must also be \(h\). Since all three have equal bases (same radius \(r\)) AND same height (\(h\)):
For the hemisphere: radius = height. So, \(r_{hemisphere} = h_{hemisphere}\).
Since all bases are equal, the radius of the cone and cylinder is also \(r\).
Since all heights are equal, the height of the cone and cylinder is also \(h\). The crucial condition is that for a hemisphere, its "height" (when standing on its base) *is* its radius. So, if they all have the same height \(h\), then for the hemisphere, its radius must be \(h\). And since they all have equal bases, the radius of the cone and cylinder must also be \(h\). Thus, for all three shapes: radius \(R = h\), and height \(H = h\). Step 2: Volume formulas
Volume of Cone (\(V_c\)): \(\frac{1}{3}\pi R^2 H\)
Volume of Hemisphere (\(V_h\)): \(\frac{2}{3}\pi R^3\)
Volume of Cylinder (\(V_{cyl}\)): \(\pi R^2 H\) Step 3: Substitute \(R=h\) and \(H=h\) into the formulas
Cone: \(V_c = \frac{1}{3}\pi (h)^2 (h) = \frac{1}{3}\pi h^3\)
Hemisphere: Its radius is \(R\). If it "stands on its base" and has height \(h\), then its radius \(R\) must be equal to \(h\). So, \(V_h = \frac{2}{3}\pi (h)^3 = \frac{2}{3}\pi h^3\)
Cylinder: \(V_{cyl} = \pi (h)^2 (h) = \pi h^3\) Step 4: Find the ratio of their volumes Ratio \(V_c : V_h : V_{cyl}\) \[ \frac{1}{3}\pi h^3 : \frac{2}{3}\pi h^3 : \pi h^3 \] We can cancel out the common factor \(\pi h^3\) from all terms (assuming \(h \neq 0\)): \[ \frac{1}{3} : \frac{2}{3} : 1 \] To get rid of the fractions, multiply all parts of the ratio by 3: \[ 3 \times \frac{1}{3} : 3 \times \frac{2}{3} : 3 \times 1 \] \[ 1 : 2 : 3 \] So, the ratio of their volumes is 1 : 2 : 3.