Find the distance along an arc on the surface of the earth that subtends a central angle of 1 minute. (1 minute = 1/60 degree) This is a nautical mile. Note that the radius of the Earth is 3960 miles and there are 1760 yards in a mile. Express your answer in both miles and yards.
how are the following relations true? S(theta + pi/2) = C(theta) and C(theta + pi/2) = -S(theta)?
To find the distance along an arc on the surface of the Earth that subtends a central angle of 1 minute, we can use the formula:
Arc length = (central angle / 360 degrees) * circumference
First, let's calculate the circumference of the Earth:
Circumference = 2 * π * radius
Given that the radius of the Earth is 3960 miles, we can substitute the value into the formula:
Circumference = 2 * π * 3960 miles
Next, let's convert the central angle from minutes to degrees:
1 minute = 1/60 degree
Now we can calculate the arc length:
Arc length = (1/60 degree / 360 degrees) * Circumference
Finally, we can convert the answer from miles to yards:
1 mile = 1760 yards
Let's plug the values into the formulas to find the answer.
To find the distance along an arc on the surface of the Earth that subtends a central angle of 1 minute, we can use the formula:
Distance = (central angle / 360 degrees) * 2 * pi * radius
Given that the central angle is 1 minute, we can convert it to degrees by dividing it by 60:
Central angle (in degrees) = 1 minute / 60 = 1/60 degree
Now, let's calculate the distance in miles:
Distance (in miles) = (1/60 degree / 360 degrees) * 2 * pi * 3960 miles ≈ 0.034906585 mile
To convert this distance to yards, multiply it by the conversion factor:
Distance (in yards) = 0.034906585 mile * 1760 yards/mile ≈ 61.32034216 yards
So, the distance along an arc on the Earth's surface that subtends a central angle of 1 minute is approximately 0.0349 miles or 61.32 yards.
s = r*theta
s = 3960 * 1/60 * pi/180 = 1.15mi
1.15mi * 1760yd/mi = 2027yd