At a miniature golf course, the fourth hole involves hitting the ball through a cylindrical tube that falls to a circular green below, as shown in the illustration. The diameter of this green is 76 inches, and the diameter of the hole is 4 inches. Using pi=3.14, determine approximately how much Astroturf is needed to cover this region.
I assume "this region" means the green, less the hole.
Since the area of a circle is pi r^2, just subtract the small area from the large area.
To determine the amount of Astroturf needed to cover the region, we first need to calculate the area of the circular green and then subtract the area of the hole.
The formula to calculate the area of a circle is: A = πr^2, where A is the area and r is the radius.
Given that the diameter of the green is 76 inches, the radius (r) can be calculated by dividing the diameter by 2:
r = 76 inches / 2 = 38 inches.
Now, we can calculate the area of the circular green by applying the formula:
A_green = π * (38 inches)^2
Using the value for pi as 3.14 and performing the calculation:
A_green = 3.14 * (38 inches)^2
A_green ≈ 3.14 * 1444 square inches
A_green ≈ 4533.76 square inches
Next, we need to calculate the area of the hole. Since the diameter of the hole is given as 4 inches, the radius of the hole (r_hole) is:
r_hole = 4 inches / 2 = 2 inches
The area of the hole can be calculated using the same formula:
A_hole = π * (2 inches)^2
A_hole ≈ 3.14 * (2 inches)^2
A_hole ≈ 3.14 * 4 square inches
A_hole ≈ 12.56 square inches
Finally, to determine the amount of Astroturf needed, we subtract the area of the hole from the area of the circular green:
Astroturf area = A_green - A_hole
Astroturf area ≈ 4533.76 square inches - 12.56 square inches
Astroturf area ≈ 4521.2 square inches (rounded to one decimal place)
Therefore, approximately 4521.2 square inches of Astroturf would be needed to cover the region.