Question

The ratio of volumes of two cones is 4:5 and the ratio of the radii of their bases is 2:3 then the ratio of their vertical height is

• A. 4:5
• B. 9:5
• C. 3:5
• D. 2:5

Answer 1

The Correct Option is B

To solve the problem, we need to determine the ratio of the vertical heights of two cones given that the ratio of their volumes is \(4:5\) and the ratio of the radii of their bases is \(2:3\).

Step 1: Formula for the Volume of a Cone The volume \(V\) of a cone is given by: \[ V = \frac{1}{3} \pi r^2 h \] where: - \(r\) is the radius of the base, - \(h\) is the height (or vertical height) of the cone. Step 2: Express the Volumes of the Two Cones Let the radii of the bases of the two cones be \(r_1\) and \(r_2\), and their heights be \(h_1\) and \(h_2\). The volumes of the two cones are: \[ V_1 = \frac{1}{3} \pi r_1^2 h_1 \] \[ V_2 = \frac{1}{3} \pi r_2^2 h_2 \] Step 3: Given Ratios We are given: 1. The ratio of the volumes: \[ \frac{V_1}{V_2} = \frac{4}{5} \] 2. The ratio of the radii: \[ \frac{r_1}{r_2} = \frac{2}{3} \] Step 4: Substitute the Ratios into the Volume Formula Using the volume formulas, we can write: \[ \frac{V_1}{V_2} = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2} = \frac{r_1^2 h_1}{r_2^2 h_2} \] Given that \(\frac{V_1}{V_2} = \frac{4}{5}\), we have: \[ \frac{r_1^2 h_1}{r_2^2 h_2} = \frac{4}{5} \] Step 5: Use the Ratio of Radii From the given ratio of radii, \(\frac{r_1}{r_2} = \frac{2}{3}\), we can square both sides to find the ratio of the squares of the radii: \[ \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{2}{3}\right)^2 = \frac{4}{9} \] Thus: \[ \frac{r_1^2}{r_2^2} = \frac{4}{9} \] Step 6: Solve for the Ratio of Heights Substitute \(\frac{r_1^2}{r_2^2} = \frac{4}{9}\) into the equation \(\frac{r_1^2 h_1}{r_2^2 h_2} = \frac{4}{5}\): \[ \frac{\frac{4}{9} h_1}{h_2} = \frac{4}{5} \] Simplify: \[ \frac{4}{9} \cdot \frac{h_1}{h_2} = \frac{4}{5} \] Multiply both sides by \(\frac{9}{4}\): \[ \frac{h_1}{h_2} = \frac{4}{5} \cdot \frac{9}{4} = \frac{9}{5} \] Final Answer: The ratio of the vertical heights of the two cones is: \[ {9:5} \]