The slant height and radius of a right circular cone are in the ratio 29 ∶ 20. If its volume is 4838.4 π cm 3, then its radius is:
24 cm
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:
\[l^2 = r^2 + h^2\]The volume \(V\) of a cone is given by the formula:
\[V = \frac{1}{3} \pi r^2 h\]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 \(h\):
\[l^2 = r^2 + h^2\]Substitute the expressions for \(l\) and \(r\) in terms of \(k\):
\[(29k)^2 = (20k)^2 + h^2\] \[841k^2 = 400k^2 + h^2\]Now, solve for \(h^2\):
\[h^2 = 841k^2 - 400k^2\] \[h^2 = 441k^2\]Taking the square root of both sides to find \(h\):
\[h = \sqrt{441k^2}\] \[h = 21k\]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:
\[V = \frac{1}{3} \pi (20k)^2 (21k)\] \[V = \frac{1}{3} \pi (400k^2) (21k)\] \[V = \frac{1}{3} \pi (8400k^3)\] \[V = 2800 \pi k^3\]Now, equate this expression for \(V\) to the given volume:
\[2800 \pi k^3 = 4838.4 \pi\]Divide both sides by \(\pi\):
\[2800 k^3 = 4838.4\]Solve for \(k^3\):
\[k^3 = \frac{4838.4}{2800}\] \[k^3 = 1.728\]To find \(k\), we take the cube root of 1.728:
\[k = \sqrt[3]{1.728}\]Since \(12^3 = 1728\), \(1.2^3 = 1.728\). Therefore,
\[k = 1.2\]Calculating the Radius
The radius was defined as \(r = 20k\). Now that we have the value of \(k\), we can calculate the radius:
\[r = 20 \times 1.2\] \[r = 24\]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:
\[V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (24)^2 (25.2) = \frac{1}{3} \pi (576) (25.2) = \frac{14515.2}{3} \pi = 4838.4 \pi\]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.
| Given Information | Formula Used | Result |
|---|---|---|
| Slant height : Radius = 29 : 20 | \(l = 29k\), \(r = 20k\) | Height \(h = 21k\) (from \(l^2 = r^2 + h^2\)) |
| Volume \(V = 4838.4 \pi \, \text{cm}^3\) | \(V = \frac{1}{3} \pi r^2 h\) | \(2800 \pi k^3 = 4838.4 \pi\) |
| Calculated \(k\) | \(k^3 = 1.728\) | \(k = 1.2\) |
| Radius \(r = 20k\) | Substitution | \(r = 20 \times 1.2 = 24\) 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.
Important Questions from Solid Figures
A cylindrical tube, open at both ends, is made of a metal sheet which is 0.5 cm thick. Its outer radius is 4 cm and length is 2 m. How much metal (in cm 3) has been used in making the tube?
The volume of a right circular cone is 308 cm 3 and the radius of its base is 7 cm. What is the curved surface area (in cm 2) of the cone? (Take π = \(\frac{22}{7} \) )
Six cubes, each of edge 2 cm, are joined end to end. What is the total surface area of the resulting cuboid in cm 2?
A solid cube of side 8 cm is dropped into a rectangular container of length 16 cm, breadth 8 cm and height 15 cm which is partly filled with water. If the cube is completely submerged, then the rise of water level (in cm) is:
The radii of two cylinders are in the ratio 3 : 4 and their heights are in the ratio 8 : 5. The ratio of their volumes is equal to:
