The figure in the image consists of a combination of geometric shapes: a rectangle, a smaller rectangle cut out at the bottom, and a semicircle at the top. To find the area of the figure, we need to calculate the areas of these individual parts and sum them up.

Step 1: Area of the rectangle

The dimensions of the main rectangle are 15 yards (height) by 9 yards (width).

$\text{Area of the rectangle} = \text{height} \times \text{width} = 15 \times 9 = 135 \, \text{square yards}$

Step 2: Area of the small cut-out rectangle

This cut-out is a smaller rectangle with dimensions 5 yards (height) by 3 yards (width).

$\text{Area of the cut-out rectangle} = 5 \times 3 = 15 \, \text{square yards}$

Step 3: Area of the semicircle

The diameter of the semicircle is 9 yards (equal to the width of the rectangle), so the radius $r$ is half of the diameter:

$r = \frac{9}{2} = 4.5 \, \text{yards}$

The area of a full circle is given by:

$\text{Area of the circle} = \pi r^2$

For a semicircle:

$\text{Area of the semicircle} = \frac{1}{2} \times \pi \times (4.5)^2 = \frac{1}{2} \times \pi \times 20.25 = 31.82 \, \text{square yards}$

Step 4: Total area of the figure

Now we sum up the areas:

$\text{Total area} = \text{Area of the rectangle} - \text{Area of the cut-out rectangle} + \text{Area of the semicircle}$

$\text{Total area} = 135 - 15 + 31.82 = 151.82 \, \text{square yards}$

Thus, the total area of the figure is approximately 151.82 square yards.

Would you like further details on any step, or do you have any questions?

Related Questions:

1. How would the area change if the semicircle were a full circle instead?
2. How can you find the perimeter of the figure?
3. What would happen to the area if the width of the rectangle was doubled?
4. How would you calculate the volume if this figure represented a 3D object, like a prism?
5. How is the area affected if the cut-out shape is triangular instead of rectangular?

Tip:

When solving composite area problems, breaking the shape down into simpler components like rectangles, triangles, or circles is a highly effective strategy!