Question

A hollow sphere whose inner and outer diameters are 4 cm and 8 cm respectively is melted to form a cone whose base is 8 cm in diameter. Find the slant height and curved surface area of the cone.

Hint:

To find the slant height of a cone, use the Pythagorean theorem, and for the curved surface area, use the formula \( \pi r l \).

Answer 1

Step 1: Calculate the volume of the hollow sphere.
The hollow sphere has inner and outer radii \( r_1 = \frac{4}{2} = 2 \, \text{cm} \) and \( r_2 = \frac{8}{2} = 4 \, \text{cm} \) respectively. The volume of the hollow sphere is the difference between the volume of the outer sphere and the volume of the inner sphere. The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Hence, the volume of the hollow sphere is: \[ V_{\text{hollow sphere}} = \frac{4}{3} \pi \left( r_2^3 - r_1^3 \right) = \frac{4}{3} \pi \left( 4^3 - 2^3 \right) \] \[ V_{\text{hollow sphere}} = \frac{4}{3} \pi \left( 64 - 8 \right) = \frac{4}{3} \pi \times 56 = \frac{224}{3} \pi \, \text{cubic cm.} \]
Step 2: Set up the volume of the cone.
The volume of the cone is given by the formula: \[ V_{\text{cone}} = \frac{1}{3} \pi r^2 h \] where \( r = \frac{8}{2} = 4 \, \text{cm} \) is the radius of the base, and \( h \) is the height of the cone. Since the volume of the cone is equal to the volume of the hollow sphere, we equate the two volumes: \[ \frac{1}{3} \pi r^2 h = \frac{224}{3} \pi \] Substituting \( r = 4 \): \[ \frac{1}{3} \pi (4)^2 h = \frac{224}{3} \pi \] Simplifying: \[ \frac{1}{3} \pi \times 16 \times h = \frac{224}{3} \pi \] Dividing both sides by \( \frac{1}{3} \pi \): \[ 16h = 224 \] \[ h = \frac{224}{16} = 14 \, \text{cm.} \]
Step 3: Find the slant height of the cone.
The slant height \( l \) of the cone can be found using the Pythagorean theorem. The radius \( r = 4 \, \text{cm} \) and the height \( h = 14 \, \text{cm} \), so: \[ l = \sqrt{r^2 + h^2} = \sqrt{4^2 + 14^2} = \sqrt{16 + 196} = \sqrt{212} = 14.56 \, \text{cm.} \]
Step 4: Find the curved surface area of the cone.
The formula for the curved surface area of a cone is: \[ A_{\text{curved}} = \pi r l \] Substituting the values of \( r = 4 \) and \( l = 14.56 \): \[ A_{\text{curved}} = \pi \times 4 \times 14.56 = 58.24 \pi \, \text{square cm.} \]
Conclusion:
The slant height of the cone is \( \boxed{14.56} \, \text{cm} \) and the curved surface area is \( \boxed{58.24 \pi} \, \text{square cm.} \)