Final answer:
The length of the arc that subtends a central angle of 800 degrees on a circle with a radius of 9 miles is 40π miles. The distance along an arc on the surface of the Earth that subtends a central angle of 7 minutes is 22π / 3 miles.
Step-by-step explanation:
A. On a circle of radius 9 miles, find the length of the arc that subtends a central angle of 800 degrees.
To find the length of the arc, we need to first convert the central angle to radians. There are 2π radians in a circle, so to convert degrees to radians, we use the formula θ (in radians) = (θ (in degrees) × π) / 180. Substituting the value of the central angle, we get θ = (800 × π) / 180 = (40π) / 9. Now, we can find the length of the arc using the formula s = rθ, where r is the radius of the circle. Substituting the value of the radius, we have s = 9 × (40π) / 9 = 40π.
B. Find the distance along an arc on the surface of the Earth that subtends a central angle of 7 minutes (1 minute = 1/60 degree). The radius of the earth is 3960 miles.
Similar to the previous question, we need to first convert the central angle to radians. Since 1 minute is equal to 1/60 degree, the central angle in radians is θ = (7/60) × π / 180. Now we can calculate the length of the arc using the formula s = rθ, where r is the radius of the Earth. Substituting the values, we have s = 3960 × (7/60) × π / 180 = 22π / 3 miles.