Now imagine a circle with a 3-mile radius to be a hypothetical metropolitan area with relatively little undeveloped land inside. If the radius of this hypothetical metropolitan area is extended to 10 miles, how much land is added to the original developed area? (Note: Use 3.1416 as the value of ð in your formula for the area of your circle. There are 640 acres in a square mile.)
The original area with a 3 mile radius was 9pi square miles
The new area with a 10 mile radius is 100pi square miles
So the new area added is 91pi square miles or
91*3.1416*640
= 182,967 acres
To find out how much land is added to the original developed area when the radius is extended, we need to calculate the difference in the areas of the two circles.
First, let's find the area of the original developed area with a 3-mile radius using the formula for the area of a circle: A = π * r^2.
A = 3.1416 * (3)^2
A ≈ 3.1416 * 9
A ≈ 28.2744 square miles
Since there are 640 acres in a square mile, we can convert this area to acres by multiplying by 640:
Area in acres = 28.2744 * 640
Area ≈ 18,095.296 acres
Now, let's find the area of the extended metropolitan area with a 10-mile radius:
A = 3.1416 * (10)^2
A ≈ 3.1416 * 100
A ≈ 314.16 square miles
Converting this area to acres, we get:
Area in acres = 314.16 * 640
Area ≈ 200,998.4 acres
To calculate the additional land added to the original developed area, subtract the original developed area from the extended metropolitan area:
Additional land = Area of extended metropolitan area - Area of original developed area
Additional land ≈ 200,998.4 acres - 18,095.296 acres
Additional land ≈ 182,903.104 acres
Therefore, when the radius is extended from 3 miles to 10 miles, approximately 182,903.104 acres of land are added to the original developed area.