Question

The base radii of two cones are in the ratio 5:3 and their heights are equal. If the volume of the first cone 750𝝅 cu centimeters, then what is the volume of the second come cu. centimeters?

• A. 250 𝝅
• B. 270 𝝅
• C. 370 𝝅
• D. 350 𝝅

Answer 1

Answer:

B. 270 𝝅

Understanding Cone Volumes and Ratios

This problem involves calculating the volume of cones and understanding how ratios affect these calculations.

Key Formula for Cone Volume:

• The volume of a cone is given by the formula: $V = \frac{1}{3}\pi r^2 h$, where '$r$' is the radius of the base and '$h$' is the height of the cone.

Given Information:

• The ratio of the base radii of two cones is 5:3. Let the radii be $r_1$ and $r_2$. So, $\frac{r_1}{r_2} = \frac{5}{3}$.

• The heights of the two cones are equal. Let the height be $h$. So, $h_1 = h_2 = h$.

• The volume of the first cone ($V_1$) is $750\pi$ cubic centimeters.

Calculating the Volume of the Second Cone:

1. Relate the volumes using the ratio of radii:
   The ratio of the volumes of the two cones can be expressed as:
   $\frac{V_1}{V_2} = \frac{\frac{1}{3}\pi r_1^2 h_1}{\frac{1}{3}\pi r_2^2 h_2}$

2. Simplify the ratio of volumes:
   Since $h_1 = h_2$, the $\frac{1}{3}\pi$ and $h$ terms cancel out:
   $\frac{V_1}{V_2} = \frac{r_1^2}{r_2^2} = \left(\frac{r_1}{r_2}\right)^2$

3. Substitute the given ratio of radii:
   We know $\frac{r_1}{r_2} = \frac{5}{3}$. Therefore,
   $\frac{V_1}{V_2} = \left(\frac{5}{3}\right)^2 = \frac{25}{9}$

4. Use the known volume of the first cone:
   We are given $V_1 = 750\pi$. Substitute this into the equation:
   $\frac{750\pi}{V_2} = \frac{25}{9}$

5. Solve for $V_2$:
   Cross-multiply to find $V_2$:
   $25 \times V_2 = 750\pi \times 9$
   $V_2 = \frac{750\pi \times 9}{25}$

6. Perform the calculation:
   $V_2 = 30\pi \times 9$
   $V_2 = 270\pi$ cubic centimeters.