The shape in the image appears to be a cylinder with two hemispheres attached to the top and bottom. To find the surface area of this shape, follow these steps:

Formula for Surface Area

The total surface area of this shape is made up of:

1. The curved surface area of the cylinder.
2. The surface area of the two hemispheres, which combine to form the surface area of a full sphere.

Step-by-Step Calculation:

1. Identify the dimensions:

   • Height of the cylindrical part: $h = 6 \, \text{ft}$
   • Diameter of the base: $d = 8 \, \text{ft}$, so the radius is $r = \frac{d}{2} = 4 \, \text{ft}$.

2. Surface area of the cylindrical part: The curved surface area of the cylinder is given by: $A_{\text{cylinder}} = 2\pi r h$ Substituting $r = 4 \, \text{ft}$ and $h = 6 \, \text{ft}$: $A_{\text{cylinder}} = 2\pi (4)(6) = 48\pi \, \text{ft}^2$

3. Surface area of the spherical part (two hemispheres): Two hemispheres combine to form a complete sphere. The surface area of a sphere is given by: $A_{\text{sphere}} = 4\pi r^2$ Substituting $r = 4 \, \text{ft}$: $A_{\text{sphere}} = 4\pi (4)^2 = 4\pi (16) = 64\pi \, \text{ft}^2$

4. Add the two areas: The total surface area is the sum of the curved surface area of the cylinder and the surface area of the sphere: $A_{\text{total}} = A_{\text{cylinder}} + A_{\text{sphere}}$ Substituting the values: $A_{\text{total}} = 48\pi + 64\pi = 112\pi \, \text{ft}^2$

5. Convert to decimal (if needed): Using $\pi \approx 3.1416$: $A_{\text{total}} = 112\pi \approx 112 \times 3.1416 = 351.86 \, \text{ft}^2$

Final Answer:

The surface area of the shape is approximately: $351.86 \, \text{ft}^2$

Do you need further clarification or details about the steps?
Here are 5 related questions to expand on this concept:

1. How do you calculate the volume of this composite shape?
2. What is the surface area of only the cylindrical part (excluding the hemispheres)?
3. How does changing the radius affect the surface area of this shape?
4. Can you derive the formula for the surface area of a sphere?
5. What is the difference between surface area and volume in 3D shapes?

Tip:

When solving for the surface area of composite shapes, break the problem into smaller parts by analyzing each component (e.g., cylinder, sphere, etc.) separately.