Calculating Cone Base Radius from Volume and Height

This problem asks us to find the radius of the base of a right circular cone given its volume and height. We will use the standard formula for the volume of a cone to solve this.

Understanding the Volume of a Cone

The volume ($\text{V}$) of a right circular cone is given by the formula:

$$V = \frac{1}{3} \pi r^2 h$$

where:

• $\text{V}$ is the volume of the cone
• $\pi$ (pi) is a mathematical constant, approximately 3.14159 or $\frac{22}{7}$
• $\text{r}$ is the radius of the base of the cone
• $\text{h}$ is the height of the cone

Applying the Formula to Find the Radius

We are given the following information:

• Volume ($\text{V}$) = 1232 cm$^3$
• Height ($\text{h}$) = 24 cm

We need to find the radius ($\text{r}$). We can rearrange the volume formula to solve for $\text{r}$:

$$V = \frac{1}{3} \pi r^2 h$$

Multiply both sides by 3:

$$3V = \pi r^2 h$$

Divide both sides by $\pi h$:

$$r^2 = \frac{3V}{\pi h}$$

Take the square root of both sides to find $\text{r}$:

$$r = \sqrt{\frac{3V}{\pi h}}$$

Calculation Steps

Now, let's substitute the given values into the rearranged formula. We will use $\pi \approx \frac{22}{7}$ for our calculation.

$$r = \sqrt{\frac{3 \times 1232}{\frac{22}{7} \times 24}}$$

To simplify, we can rewrite the expression under the square root:

$$r = \sqrt{\frac{3 \times 1232 \times 7}{22 \times 24}}$$

Let's perform the calculations:

• First, simplify 3 and 24: $\frac{3}{24} = \frac{1}{8}$
• The expression becomes: $\sqrt{\frac{1232 \times 7}{22 \times 8}}$
• Next, simplify 1232 and 22: $\frac{1232}{22} = 56$
• The expression becomes: $\sqrt{\frac{56 \times 7}{8}}$
• Finally, simplify 56 and 8: $\frac{56}{8} = 7$
• The expression becomes: $\sqrt{7 \times 7}$

$$r = \sqrt{49}$$

$$r = 7 \text{ cm}$$

Thus, the radius of the base of the right circular cone is 7 cm.

Revision Table: Cone Formulas

Additional Information: Properties of Right Circular Cone

A right circular cone is a three-dimensional geometric shape that tapers smoothly from a flat, circular base to a point called the apex or vertex. The axis of the cone, which connects the apex to the center of the base, is perpendicular to the base. Key properties include:

• Base: A single circular base.
• Apex: A single vertex opposite the base.
• Height ($\text{h}$): The perpendicular distance from the apex to the center of the base.
• Radius ($\text{r}$): The radius of the circular base.
• Slant Height ($\text{l}$): The distance from the apex to any point on the circumference of the base. It forms the hypotenuse of a right triangle with the height and radius as the other two sides ($l^2 = r^2 + h^2$).
• Cross Section: A cross section parallel to the base is a circle. A cross section through the apex is a triangle.

Understanding these properties is crucial for solving problems involving the volume, surface area, and dimensions of a cone.