The area of the base of a cone is 144π cm 2while its slant height is 13 cm. This cone is remoulded to obtain a solid sphere. The radius of this sphere will be-
∛180 cm
Solving the Cone to Sphere Remoulding Problem
The question asks us to find the radius of a solid sphere obtained by remoulding a cone with a specific base area and slant height. The key principle here is that when a solid is remoulded into another shape, its volume remains constant.
Calculating Cone Dimensions
We are given the base area of the cone and its slant height. We need to find the radius of the base and the height of the cone to calculate its volume.
- Base Area of Cone: The base of a cone is a circle. The area of the circular base is given as $144\pi$ cm${}^2$. The formula for the area of a circle is $\pi r^2$, where $r$ is the radius of the base.
\(\pi r^2 = 144\pi\)
Dividing both sides by \(\pi\), we get:
\(r^2 = 144\)
Taking the square root of both sides to find the radius of the base:
\(r = \sqrt{144} = 12\) cm
- Slant Height of Cone: The slant height (\(l\)) is given as 13 cm.
- Height of Cone: We can find the height (\(h\)) of the cone using the Pythagorean theorem, which relates the base radius, height, and slant height of a cone: \(l^2 = r^2 + h^2\).
\(13^2 = 12^2 + h^2\)
\(169 = 144 + h^2\)
\(h^2 = 169 - 144\)
\(h^2 = 25\)
Taking the square root to find the height:
\(h = \sqrt{25} = 5\) cm
Calculating the Volume of the Cone
Now that we have the base radius (\(r\)) and the height (\(h\)) of the cone, we can calculate its volume using the formula for the volume of a cone: Volume${}_{\text{cone}} = \frac{1}{3}\pi r^2 h$.
\(\text{Volume}_{\text{cone}} = \frac{1}{3} \pi (12 \text{ cm})^2 (5 \text{ cm})\)
\(\text{Volume}_{\text{cone}} = \frac{1}{3} \pi (144 \text{ cm}^2) (5 \text{ cm})\)
\(\text{Volume}_{\text{cone}} = \frac{1}{3} \pi (720 \text{ cm}^3)\)
\(\text{Volume}_{\text{cone}} = 240\pi \text{ cm}^3\)
Calculating the Radius of the Sphere
When the cone is remoulded into a solid sphere, the volume remains the same. Let \(R\) be the radius of the sphere. The formula for the volume of a sphere is Volume${}_{\text{sphere}} = \frac{4}{3}\pi R^3$.
Since the volume is conserved:
\(\text{Volume}_{\text{sphere}} = \text{Volume}_{\text{cone}}\)
\(\frac{4}{3}\pi R^3 = 240\pi\)
To solve for \(R\), first divide both sides by \(\pi\):
\(\frac{4}{3} R^3 = 240\)
Multiply both sides by \(\frac{3}{4}\):
\(R^3 = 240 \times \frac{3}{4}\)
\(R^3 = (60 \times 4) \times \frac{3}{4}\)
\(R^3 = 60 \times 3\)
\(R^3 = 180\)
To find \(R\), take the cube root of both sides:
\(R = \sqrt[3]{180}\) cm
Thus, the radius of the solid sphere is \(\sqrt[3]{180}\) cm.
| Shape | Formula | Value |
|---|---|---|
| Cone Base Area | \(\pi r^2\) | \(144\pi\) cm\({}^2\) |
| Cone Base Radius (\(r\)) | \(\sqrt{\text{Area}/\pi}\) | 12 cm |
| Cone Slant Height (\(l\)) | Given | 13 cm |
| Cone Height (\(h\)) | \(\sqrt{l^2 - r^2}\) | 5 cm |
| Cone Volume | \(\frac{1}{3}\pi r^2 h\) | \(240\pi\) cm\({}^3\) |
| Sphere Volume | \(\frac{4}{3}\pi R^3\) | \(\frac{4}{3}\pi R^3\) |
| Sphere Radius (\(R\)) | \(\sqrt[3]{\frac{3 \times \text{Volume}}{4\pi}}\) | \(\sqrt[3]{180}\) cm |
Conclusion on Sphere Radius
The radius of the solid sphere obtained by remoulding the cone is \(\sqrt[3]{180}\) cm. This matches one of the given options.
Revision Table: Geometric Volume Formulas
| Shape | Formula for Volume | Key Variables |
|---|---|---|
| Cone | \(\frac{1}{3}\pi r^2 h\) | \(r\): base radius, \(h\): height |
| Cylinder | \(\pi r^2 h\) | \(r\): base radius, \(h\): height |
| Sphere | \(\frac{4}{3}\pi R^3\) | \(R\): radius |
| Cube | \(s^3\) | \(s\): side length |
| Cuboid | \(lwh\) | \(l\): length, \(w\): width, \(h\): height |
Additional Information: Conservation of Volume
The principle of conservation of volume is fundamental in problems involving melting, casting, or remoulding of solids. When a solid material is transformed from one shape to another without any loss or addition of material, the total amount of space it occupies (its volume) remains constant.
In this problem, the metal (or material) of the cone is used to form the sphere. No material is lost or added during this process. Therefore, the volume of the original cone is equal to the volume of the resulting sphere. This principle allows us to equate the volume formulas of the two shapes and solve for the unknown dimension, which was the radius of the sphere in this case.
Understanding this concept is crucial for solving many problems related to the volumes of 3D shapes where transformations occur.
Similar Questions
A solid sphere of diameter 12 cm is melted and three shorts are prepared. If the diameters of two shorts are 6 cm and 10 cm respectively, what is the surface area (in cm 2) of the third short?
The volume of a right circular cone, whose radius of the base is same as one-third of its altitude, and the volume of a sphere are equal. The ratio of the radius of the cone to the radius of the sphere is:
The surface area of three faces of a cuboid sharing a vertex are 20 m 2, 32 m 2and 40 m 2. What is the volume of the cuboid?
The total surface area of a cuboid is 236 cm 2. Its length is 8 cm and height is 6 cm. Find its breadth (in cm).
A shuttle cock used for playing badminton has the shape of a frustum of a cone mounted on a hemisphere. The two diameters of the frustum are 5 cm and 2 cm, the height of the entire shuttle cock is 6 cm. Find the external surface area.
If the edge of a cube is increased by 2 cm, the volume will increase by 488 cm3. What then will be the length of each edge of the cube?
The area of the base of a cone is 64π cm 2while its slant height is 17 cm. This cone is remoulded to obtain a solid sphere. Find the radius of this sphere.
The volume of a right circular cone, whose radius of the base is half of its altitude, and the volume of a hemisphere are equal. The ratio of the radius of the cone to the radius of the hemisphere is:
From a solid cube of side 7 cm, a conical cavity of height 7 cm and radius 3 cm is hollowed out. Find the volume of remaining solid.
A sphere is split in the ratio 1 : 3. The larger part is moulded into a cone having a height equal to the radius of its base, while the smaller part is moulded into a cylinder having a height equal to the radius of its base. What would be the ratio of the radius of the cone to the height of the cylinder?
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 = \(\frac{22}{7} \) )
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:
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:
