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.

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

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.