Question

The ratio of the radii of two cylinders is 4 : 5 and the ratio of their heights is 6 : 7, then the ratio of their volumes is

• A. 96 : 125
• B. 96 : 175
• C. 175 : 96
• D. 20 : 63

Hint:

Remember that the volume of a cylinder depends on the square of the radius but only on the first power of the height. So, you need to square the ratio of the radii before multiplying by the ratio of the heights.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to find the ratio of the volumes of two cylinders given the ratios of their radii and heights.

Step 2: Key Formula or Approach:
The formula for the volume of a cylinder is \(V = \pi r^2 h\).
The ratio of the volumes of two cylinders will be:
\[ \frac{V_1}{V_2} = \frac{\pi r_1^2 h_1}{\pi r_2^2 h_2} = \left(\frac{r_1}{r_2}\right)^2 \times \left(\frac{h_1}{h_2}\right) \]

Step 3: Detailed Explanation:
We are given the ratios:
Ratio of radii: \(\frac{r_1}{r_2} = \frac{4}{5}\)
Ratio of heights: \(\frac{h_1}{h_2} = \frac{6}{7}\)
Now, substitute these ratios into the volume ratio formula:
\[ \frac{V_1}{V_2} = \left(\frac{4}{5}\right)^2 \times \left(\frac{6}{7}\right) \] \[ \frac{V_1}{V_2} = \frac{16}{25} \times \frac{6}{7} \] \[ \frac{V_1}{V_2} = \frac{16 \times 6}{25 \times 7} = \frac{96}{175} \] The ratio of the volumes is 96 : 175.

Step 4: Final Answer:
The ratio of their volumes is 96 : 175.