To solve this problem, we need to find the surface area of a figure that resembles a hemisphere (top half of a sphere) resting on top of a cylinder.

Step-by-Step Solution:

1. Identify the dimensions:

   • The diameter of both the hemisphere and the cylinder is $9$ yards, which means the radius $r$ is: $r = \frac{9}{2} = 4.5 \text{ yards}$
   • The height of the cylinder is given as $3$ yards.

2. Surface Area of the Hemisphere:

   • The surface area of a hemisphere (not including the circular base) is given by: $\text{Surface Area of Hemisphere} = 2 \pi r^2$
   • Using $\pi \approx 3.14$ and $r = 4.5$: $\text{Surface Area of Hemisphere} = 2 \times 3.14 \times (4.5)^2$ $= 2 \times 3.14 \times 20.25$ $= 127.17 \text{ square yards}$

3. Surface Area of the Cylinder (curved part only):

   • The lateral surface area of a cylinder is given by: $\text{Lateral Surface Area of Cylinder} = 2 \pi r h$
   • With $r = 4.5$ and $h = 3$: $\text{Lateral Surface Area of Cylinder} = 2 \times 3.14 \times 4.5 \times 3$ $= 2 \times 3.14 \times 13.5$ $= 84.78 \text{ square yards}$

4. Total Surface Area:

   • To find the total surface area, we add the surface area of the hemisphere and the lateral surface area of the cylinder (note that the base of the hemisphere and the top of the cylinder are not included since they are internal): $\text{Total Surface Area} = 127.17 + 84.78$ $= 211.95 \text{ square yards}$

Final Answer:

The surface area of the figure is approximately 211.95 square yards.

Would you like more details or have any questions?

Related Questions to Explore:

1. How would the surface area change if the radius was doubled?
2. What would be the volume of this combined hemisphere and cylinder?
3. How would the calculation change if the height of the cylinder was increased to 6 yards?
4. What is the formula for the surface area of a full sphere?
5. How can you find the surface area if given the circumference instead of the diameter?

Tip:

For composite shapes, break down the surface area calculation by focusing on each part individually and then sum them for the total area.