Solution (Question 1):
We can see that both circles have the same radius. This means that the circles have same areas. Therefore, the area of the figure is the difference of the area of the rectangle and the sum of the area of the two circles.
Area of rectangle:
The area of a rectangle is the vertical side length multiplied with the horizontal side length.
- Area of rectangle = Length × Width
- Area of rectangle = 12 × 6
- Area of rectangle = 72 ft²
Area of circle:
Substitute the radius in the formula to determine the area of the circle.
- Area of circle = πr²
- Area of circle = π(3)²
- Area of circle = π(9)
- Area of circle ≈ (3.14)(9)
- Area of circle ≈ 28.26 ft²
Area of shaded region:
- Area of shaded region ≈ Area (rectangle) - [Area (circle) + Area (circle)]
- Area of shaded region ≈ 72 - (28.26 + 28.26)
- Area of shaded region ≈ 15.48 ft² ≈ 16 ft² (Option E)
Solution (Question 2):
To determine the area of the shaded region, we need to determine the area of the square, then subtract it with the area of the circle.
Area of square:
The area of a square is the measure of its side, squared. Therefore,
- Area of square = (Side)²
- Area of square = (16)²
- Area of square = 256 in²
Area of circle:
The formula that determines the area of a circle is πr². We can tell that the radius of the circle is required to determine the area of the circle, as the formula to determine the area of a circle (πr²) includes the variable "r", which is the radius of the circle.
How to determine the radius of the circle?
Since we are given the diameter of the circle, we can determine the radius of the circle by creating an equation that represents the relationship between a diameter and radius. Then, substitute the diameter in the equation and solve for the radius of the circle.
Determining the radius of the circle:
It should be noted that the diameter is twice larger than the radius. Therefore, the equation that represents the relationship between the radius of the circle and the diameter of the circle is;
Substitute the diameter in the equation to determine the radius of the circle.
Divide 2 to both sides of the equation and simplify both sides:
- 16 inches = 2(Radius)
- 16 inches/2 = 2(Radius)/2
- Radius = 8 inches
Substitute the radius in the formula to determine the area of the circle.
- Area of circle = πr²
- Area of circle = π(8)²
- Area of circle = π(64)
Take π as 3.14
- Area of circle = (3.14)(64)
- Area of circle ≈ 200.96 inches²
Area of shaded region:
- Area of shaded region ≈ Area of square - Area of circle
- Area of shaded region ≈ 256 - 200.96
- Area of shaded region ≈ 55.04 inches² ≈ 55 inches² (Option C)
Final answers: