Given Values
- Radius of the sphere (r) = 5 cm
- Radius of the cone (R) = unknown
- Height of the cone (h) = unknown
Surface Area of the Sphere
- The formula for the surface area of a sphere is:
- Surface Area = 4 * π * r^2
- Substituting the radius:
- Surface Area = 4 * π * (5)^2
- Surface Area = 4 * π * 25 = 100π cm²
Curved Surface Area of the Cone
- The formula for the curved surface area of a cone is:
- Curved Surface Area = π * R * l
- Where l is the slant height of the cone, calculated using the Pythagorean theorem:
- l = √(R² + h²)
Relation Between Areas
- Given that the surface area of the sphere is 5 times the curved surface area of the cone:
- 100π = 5 * (π * R * l)
- Simplifying gives:
- 100 = 5 * R * l
- 20 = R * l
Finding the Height of the Cone
- Using the expression for l:
- l = √(R² + h²)
- Substitute into the equation:
- 20 = R * √(R² + h²)
Rearranging gives:
- 20^2 = R² * (R² + h²)
- 400 = R^4 + R²h²
- Now, if we choose a radius for the cone, we can solve for h.
Example Calculation
- Let’s assume R = 4 cm:
- 400 = (4)^4 + (4²)h²
- 400 = 256 + 16h²
- 16h² = 144
- h² = 9
- h = 3 cm
Conclusion
- The height of the cone can be determined based on the chosen radius. For R = 4 cm, the height is 3 cm. Adjust R as needed to find other possible heights.