This question asks us to calculate the total area that will be painted blue, which is the combined area of the rectangle and the semicircle in the diagram.

Step-by-Step Solution:

1. Area of the Rectangle: The rectangle's dimensions are given as:

   • Length = $15 \, \text{ft}$
   • Width = $10 \, \text{ft}$

   The area of a rectangle is calculated as: $\text{Area} = \text{Length} \times \text{Width}$

   Substituting the values: $\text{Area of Rectangle} = 15 \times 10 = 150 \, \text{sq ft}$

2. Area of the Semicircle: The semicircle has a diameter equal to the width of the rectangle, which is $10 \, \text{ft}$. Therefore, the radius ($r$) is: $r = \frac{\text{Diameter}}{2} = \frac{10}{2} = 5 \, \text{ft}$

   The area of a semicircle is calculated as half the area of a circle: $\text{Area of Semicircle} = \frac{1}{2} \pi r^2$

   Substituting $r = 5$: $\text{Area of Semicircle} = \frac{1}{2} \pi (5)^2 = \frac{1}{2} \pi (25) = 12.5 \pi \, \text{sq ft}$

   Using $\pi \approx 3.14$: $\text{Area of Semicircle} \approx 12.5 \times 3.14 = 39.25 \, \text{sq ft}$

3. Total Area to be Painted: The total area is the sum of the rectangle and the semicircle: $\text{Total Area} = \text{Area of Rectangle} + \text{Area of Semicircle}$

   Substituting the values: $\text{Total Area} = 150 + 39.25 = 189.25 \, \text{sq ft}$

Final Answer:

The total area to be painted blue is approximately: $\boxed{189.25 \, \text{square feet}}$

Do you want further clarifications or additional details? Here are 5 related questions for practice:

1. What is the area of a full circle with a radius of $5 \, \text{ft}$?
2. If the dimensions of the rectangle were doubled, what would the new painted area be?
3. How would the calculation change if the semicircle were replaced by a full circle?
4. What is the perimeter of the entire painted region (rectangle + semicircle)?
5. How would the area change if the width of the rectangle were $12 \, \text{ft}$ instead of $10 \, \text{ft}$?

Tip: Always ensure that units are consistent throughout your calculations to avoid errors.