The problem asks us to find the radius of a right circular cone given the ratio of its slant height to radius and its volume. We are given that the ratio of the slant height \(l\) to the radius \(r\) is 29 ∶ 20, and the volume \(V\) of the cone is \(4838.4 \pi \, \text{cm}^3\).

Understanding Cone Geometry and Formulas

A right circular cone has a circular base and an apex directly above the center of the base. The slant height \(l\), radius \(r\), and height \(h\) of a right cone form a right-angled triangle, with the slant height as the hypotenuse. The relationship is given by the Pythagorean theorem:

The volume \(V\) of a cone is given by the formula:

Setting up the Variables

The ratio of slant height to radius is given as \(l:r = 29:20\). We can represent the slant height and radius using a constant \(k\):

• Slant height, \(l = 29k\)
• Radius, \(r = 20k\)

Here, \(k\) is a positive constant.

Calculating the Height of the Cone

We need the height \(h\) to use the volume formula. We can find \(h\) using the Pythagorean theorem relating \(l\), \(r\), and \(h\):

Substitute the expressions for \(l\) and \(r\) in terms of \(k\):

Now, solve for \(h^2\):

Taking the square root of both sides to find \(h\):

So, the height of the cone is \(21k\).

Using the Volume to Find the Constant \(k\)

The volume of the cone is given as \(V = 4838.4 \pi \, \text{cm}^3\). We also have the volume formula \(V = \frac{1}{3} \pi r^2 h\). Substitute the expressions for \(r\) and \(h\) in terms of \(k\) into the volume formula:

Now, equate this expression for \(V\) to the given volume:

Divide both sides by \(\pi\):

Solve for \(k^3\):

To find \(k\), we take the cube root of 1.728:

Since \(12^3 = 1728\), \(1.2^3 = 1.728\). Therefore,

Calculating the Radius

The radius was defined as \(r = 20k\). Now that we have the value of \(k\), we can calculate the radius:

The radius of the cone is 24 cm.

Verification

Let's check if this radius value is consistent with the given volume. If \(r = 24\), then \(k = r/20 = 24/20 = 1.2\). The height is \(h = 21k = 21 \times 1.2 = 25.2\) cm. The slant height is \(l = 29k = 29 \times 1.2 = 34.8\) cm. Let's check \(l^2 = r^2 + h^2\): \(34.8^2 = 1211.04\) and \(24^2 + 25.2^2 = 576 + 635.04 = 1211.04\). The dimensions are consistent. Now let's calculate the volume:

This matches the given volume. So, the radius of 24 cm is correct.

Conclusion

Based on the given ratio of slant height to radius and the volume of the right circular cone, we found the radius to be 24 cm.

Revision Table: Cone Formulas

Here are some key formulas related to a right circular cone:

• Volume \(V = \frac{1}{3} \pi r^2 h\)
• Curved Surface Area \(A_{CSA} = \pi r l\)
• Total Surface Area \(A_{TSA} = \pi r (r+l)\)
• Relationship between \(l\), \(r\), and \(h\): \(l^2 = r^2 + h^2\) (Pythagorean Theorem)

Additional Information: Right Circular Cone Properties

A right circular cone is a cone where the apex is directly above the center of the circular base. This specific geometry allows for the height, radius, and slant height to form a right triangle, which is crucial for solving problems like this one using the Pythagorean theorem. The ratio of dimensions in geometry problems often simplifies calculations by introducing a common factor, as seen with the constant \(k\) in this solution.