If two points A and B are d miles apart, express angle ACB in radians and in degrees. (The diameter of Earth is approximately 8000 miles. Round your answer to the nearest minute.)
d = 700
Is C the center of the earth?
If so, you have an arc of a great circle, so
d = rθ
To find the angle ACB, we need to use trigonometry. First, let's draw a diagram to visualize the situation:
A C B
In this diagram, A and B represent the two points that are d miles apart, and C is the center of the Earth. The angle ACB represents the angle subtended by the arc between points A and B.
To find angle ACB in radians, we can use the formula:
angle_in_radians = arc_length / radius_of_circle
In this case, the arc length is the distance between points A and B (d), and the radius of the circle is the radius of the Earth (8000 miles).
angle_in_radians = d / radius_of_circle
= 700 / 8000
Calculating this, we get:
angle_in_radians = 0.0875 radians
To convert this to degrees, we can use the fact that 1 radian is equal to 180 degrees divided by pi:
angle_in_degrees = angle_in_radians * (180 / pi)
= 0.0875 * (180 / pi)
Now, let's calculate this value:
angle_in_degrees = 0.0875 * (180 / 3.14159)
≈ 5.001 degrees
Rounding this to the nearest minute, we have:
angle_in_degrees ≈ 5 degrees