Let's solve the problem step by step.

Given:

• Sphere radius $r = 6$ m.
• Cone base diameter = 8 m, so cone radius $r = 4$ m.
• The surface area of the sphere is equal to the curved surface area of the cone.

Step 1: Find the Surface Area of the Sphere

The formula for the surface area of a sphere is: $A = 4\pi r^2$ Substituting $r = 6$:

$A = 4\pi (6^2) = 144\pi \text{ m}^2$

Step 2: Find the Curved Surface Area of the Cone

The curved surface area of a cone is given by:

$A = \pi r l$

where $r = 4$ m and $l$ is the slant height.

Since this is equal to the sphere’s surface area:

$\pi (4) l = 144\pi$

Cancel $\pi$ on both sides:

$4l = 144$

Step 3: Solve for $l$

$l = \frac{144}{4} = 36 \text{ m}$

Final Answer:

The slant height of the cone is 36.0 m (to 1 decimal place).

Would you like a breakdown of any step? Here are some related questions you might find interesting:

1. How do we derive the formula for the surface area of a sphere?
2. What is the full surface area (including the base) of the cone?
3. How do we calculate the volume of the cone and the sphere?
4. How does changing the radius of the cone affect the slant height?
5. What real-life applications involve using curved surface areas?

Tip: Always check if given dimensions are radii or diameters to avoid calculation mistakes!