To solve the problem:

1. Surface Area of Sphere: The surface area of a sphere is given by: $A_{\text{sphere}} = 4\pi r^2$ Here, $r = 5 \, \text{cm}$: $A_{\text{sphere}} = 4\pi (5)^2 = 100\pi \, \text{cm}^2$

2. Curved Surface Area of Cone: Denote the curved surface area of the cone by $A_{\text{cone}}$. According to the problem: $A_{\text{sphere}} = 5 \cdot A_{\text{cone}}$ Substituting $A_{\text{sphere}} = 100\pi$: $100\pi = 5 \cdot A_{\text{cone}} \implies A_{\text{cone}} = 20\pi \, \text{cm}^2$

   The formula for the curved surface area of a cone is: $A_{\text{cone}} = \pi r l$ where $r$ is the radius and $l$ is the slant height. Substituting $r = 4 \, \text{cm}$ and $A_{\text{cone}} = 20\pi$: $20\pi = \pi (4) l \implies l = 5 \, \text{cm}$

3. Height of the Cone: Using the Pythagorean theorem: $l^2 = r^2 + h^2$ Substituting $l = 5 \, \text{cm}$ and $r = 4 \, \text{cm}$: $(5)^2 = (4)^2 + h^2 \implies 25 = 16 + h^2 \implies h^2 = 9 \implies h = 3 \, \text{cm}$

4. Volume of the Cone: The formula for the volume of a cone is: $V = \frac{1}{3} \pi r^2 h$ Substituting $r = 4 \, \text{cm}$ and $h = 3 \, \text{cm}$: $V = \frac{1}{3} \pi (4)^2 (3) = \frac{1}{3} \pi (16) (3) = 16\pi \, \text{cm}^3$

Final Answers:

• Height of the cone: $3 \, \text{cm}$
• Volume of the cone: $16\pi \, \text{cm}^3$

Would you like detailed explanations or diagrams for this solution?

Related Questions:

1. How would the answer change if the sphere's radius was doubled?
2. What happens to the volume of the cone if its height is halved?
3. How is the relationship between slant height and base radius derived in cones?
4. What is the general formula for the curved surface area of a cone in terms of height?
5. How do the formulas for spheres and cones relate in terms of $\pi$?

Tip: Always verify your calculations by substituting back into the original conditions!