Question

If the radii of two cylinders are in the ratio \(2:3\) and their height are in the ratio \(5:3\). The ratio of their volumes is :

• A. \(10:17\)
• B. \(20:27\)
• C. \(17:27\)
• D. \(20:37\)

Hint:

1. Volume of cylinder: \(V = \pi r^2 h\). 2. Ratio of volumes: \(\frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right)\). 3. Given \(\frac{r_1}{r_2} = \frac{2}{3}\) and \(\frac{h_1}{h_2} = \frac{5}{3}\). 4. Substitute: \(\frac{V_1}{V_2} = \left(\frac{2}{3}\right)^2 \times \frac{5}{3} = \frac{4}{9} \times \frac{5}{3} = \frac{20}{27}\). 5. The ratio is \(20:27\).

Answer 1

The Correct Option is B

Concept: The volume (\(V\)) of a cylinder is given by the formula \(V = \pi r^2 h\), where \(r\) is the radius of the base and \(h\) is the height of the cylinder. Step 1: Define the ratios for the two cylinders Let the first cylinder be Cylinder 1 and the second cylinder be Cylinder 2. Let their radii be \(r_1\) and \(r_2\), their heights be \(h_1\) and \(h_2\), and their volumes be \(V_1\) and \(V_2\). Given: Ratio of radii: \(\frac{r_1}{r_2} = \frac{2}{3}\) This means we can write \(r_1 = 2k\) and \(r_2 = 3k\) for some constant \(k\). Ratio of heights: \(\frac{h_1}{h_2} = \frac{5}{3}\) This means we can write \(h_1 = 5j\) and \(h_2 = 3j\) for some constant \(j\). Step 2: Write the formulas for the volumes of the two cylinders Volume of Cylinder 1: \(V_1 = \pi r_1^2 h_1\) Volume of Cylinder 2: \(V_2 = \pi r_2^2 h_2\) Step 3: Find the ratio of their volumes We need to find \(\frac{V_1}{V_2}\): \[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} \] The \(\pi\) terms cancel out: \[ \frac{V_1}{V_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \] This can be written as: \[ \frac{V_1}{V_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right) \] Step 4: Substitute the given ratios into the expression Substitute \(\frac{r_1}{r_2} = \frac{2}{3}\) and \(\frac{h_1}{h_2} = \frac{5}{3}\): \[ \frac{V_1}{V_2} = \left(\frac{2}{3}\right)^2 \times \left(\frac{5}{3}\right) \] \[ \frac{V_1}{V_2} = \left(\frac{2^2}{3^2}\right) \times \frac{5}{3} \] \[ \frac{V_1}{V_2} = \frac{4}{9} \times \frac{5}{3} \] Step 5: Calculate the final ratio \[ \frac{V_1}{V_2} = \frac{4 \times 5}{9 \times 3} = \frac{20}{27} \] So, the ratio of their volumes \(V_1 : V_2\) is \(20:27\). This matches option (2).