The boundary of a park is shaped like a circle. The park has a rectangular playground in the center and 2 square flower beds, one on each side of the playground. The length of the playground is l and its width is w. The length of each side of the flower beds is a. Which two equivalent expressions represent the total fencing material required to surround the playground and flower beds? Assume that the playground and beds do not overlap.
The circumference of the circle is 2πr, where r is the radius of the circle. Since the park is shaped like a circle, the circumference represents the total fencing material required to surround the park.
The radius of the circle is equal to half the length of the playground, or l/2.
So, the total fencing material required to surround the park is 2π(l/2) = πl.
Alternatively, the total fencing material required to surround the park is also equal to the sum of the perimeters of the rectangular playground and the two square flower beds.
The perimeter of the rectangular playground is equal to 2(l + w).
Since there are two square flower beds, each with a side length of a, the total perimeter of the flower beds is equal to 4a.
So, the total fencing material required to surround the park is 2(l + w) + 4a.
To calculate the total fencing material required to surround the playground and flower beds, we need to find the perimeter of the rectangle and the perimeters of the two squares.
The rectangle has a length of l and a width of w, so its perimeter is given by the formula:
P_rectangle = 2l + 2w
The two squares have sides of length a, so each square's perimeter is given by the formula:
P_square = 4a
To calculate the total fencing material required, we add the perimeters of the rectangle and the two squares:
P_total = P_rectangle + 2(P_square)
Substituting the formulas for P_rectangle and P_square, we get:
P_total = 2l + 2w + 2(4a)
This expression simplifies to:
P_total = 2l + 2w + 8a
Therefore, the two equivalent expressions that represent the total fencing material required are:
2l + 2w + 8a and P_total.
To find the total fencing material required to surround the playground and the flower beds, we need to calculate the perimeter of each of these shapes.
First, let's calculate the perimeter of the playground, which is a rectangle.
A rectangle has two sides of length l and two sides of width w. Therefore, the perimeter of the playground is given by the formula:
Perimeter of the playground = 2l + 2w.
Next, let's calculate the perimeter of one flower bed, which is a square.
A square has all sides of equal length, a. Therefore, the perimeter of one flower bed is given by the formula:
Perimeter of one flower bed = 4a.
Since there are two flower beds, we need to calculate the total perimeter of both flower beds. It is given by:
Perimeter of both flower beds = 2 * (Perimeter of one flower bed) = 2 * 4a = 8a.
Now, to find the total fencing material required to surround the playground and the flower beds, we need to add the perimeters of the playground and the flower beds.
Total fencing material required = Perimeter of the playground + Perimeter of both flower beds.
This can be expressed as:
Total fencing material required = (2l + 2w) + (8a).
Therefore, the two equivalent expressions that represent the total fencing material required to surround the playground and flower beds are:
1) 2l + 2w + 8a
2) 8a + 2l + 2w