Astronauts on the Moon look back at Earth, a distance of about 400,000 km away. If the cities of Washington, D.C. and New York are separated by 300 km, what will be the angle between them when viewed from the Moon?
To find the angle between Washington, D.C. and New York when viewed from the Moon, we can use trigonometry.
First, let's consider a triangle formed by the Moon, Washington, D.C., and New York. The distance between Washington, D.C. and New York is given as 300 km. But since we are looking at it from the Moon, we need to consider the distance from the Moon to each of these cities.
The distance from the Moon to Earth is about 400,000 km.
Now, let's consider the angle at the Moon. We can use the Law of Cosines to find this angle:
c^2 = a^2 + b^2 - 2ab * cos(C)
Where:
- c is the distance from the Moon to Washington, D.C. (400,000 km)
- a is the distance from the Moon to New York (400,000 km)
- b is the distance between Washington, D.C. and New York (300 km)
- C is the angle at the Moon (the angle between Washington, D.C. and New York)
Plugging in the values, we get:
(400,000 km)^2 = (400,000 km)^2 + (300 km)^2 - 2(400,000 km)(400,000 km) * cos(C)
Simplifying the equation, we get:
(400,000 km)^2 - (400,000 km)^2 - (300 km)^2 = - 2(400,000 km)(400,000 km) * cos(C)
(400,000 km)^2 - (400,000 km)^2 - (300 km)^2 = - 2(400,000 km)^2 * cos(C)
(300 km)^2 = 2(400,000 km)^2 * cos(C)
Now, we can solve for cos(C) by rearranging the equation:
cos(C) = (300 km)^2 / (2(400,000 km)^2)
cos(C) = 0.0000225
Finally, we can find the angle C by taking the inverse cosine (cos^-1) of cos(C):
C ≈ cos^-1(0.0000225)
Using a calculator, we find:
C ≈ 89.9865 degrees
Therefore, the angle between Washington, D.C. and New York when viewed from the Moon is approximately 89.9865 degrees.