The Moon has a diameter of about 3480 km and an orbital radius of about 384 400 km from the centre of Earth. Suppose that the Moon is directly overhead. What is the measure of the angle subtended by the diameter to the Moon as measured by an astronomer on the surface of Earth? Answer in both radians and degrees.
The other answer didn't really help me at all so I hope my explanation works better.
What we're trying to find is the angle subtended from the diameter of the moon right? That's the angle that's right opposite the side-length aka diameter of the moon if you need to rephrase that in your head to make sense.
We'll call that angle ø. Hopefully you already know that how to find angle ø but if you don't —> ø = a/r.
a means arc-length
r means radius
So the question has really just handed us the answer.
The earth is where angle ø is. The orbital radius is just the radius. The diameter of the moon is our arc length. You don't need to know anything about the radius of the earth because it's not given to you, and the question isn't asking for you to find that.
Let's go over it again.
ø = a/r
a (arc length) = diameter = 3480 km
r (radius) = orbital radius = 384400 km
thus ø = 3480/384400 (MAKE SURE YOU PUT IT INTO YOUR CALCULATOR RIGHT THE FIRST TIME. That was what kept me stuck on this lol)
ø = approx. 0.009 radian.
Which in degrees is
0.009 rad x 180/π = 0.5˚ approx.
There you go.
Sorry, I actually have a question. According to the question that was posted. If I were to solve this question, without knowing the radius of the earth before hand. Is there any other solution??
Diameter of the moon = 3480 km
Distance from surface of the earth
= Distance to centre - radius of the earth
= 384400-6400 km
= 378000 km
Since the angle is very small, we can approximate the arc-length by the chord length, equal to the diameter.
Angle subtended
= 3480/378000
= 0.0092 radian
If we had used the chord instead, the difference in angle calculated would have been 0.0000000325 radians more.
i.e.
angle
= 2*sin-1(1740/37800)
= 0.0092 radian
Well, if the Moon is directly overhead, it means that the angle subtended by the diameter to the Moon is a right angle, since the line connecting the observer's eye, the center of the Earth, and the Moon's diameter form a right triangle.
Now, to calculate the measure of the angle in radians, we can use the formula:
Angle in radians = Distance / Radius
The diameter of the Moon is 3480 km, so the distance between the observer and the Moon's diameter is also 3480 km. The orbital radius of the Moon from the center of the Earth is 384,400 km.
So, the angle in radians would be:
Angle in radians = 3480 km / 384,400 km ≈ 0.0090484 radians
To convert this to degrees, we can use the fact that 1 radian is approximately equal to 57.3 degrees.
Therefore, the angle in degrees would be:
Angle in degrees = 0.0090484 radians x 57.3 degrees/radian ≈ 0.5185 degrees
So, the angle subtended by the diameter to the Moon as measured by an astronomer on the surface of Earth is approximately 0.0090484 radians or 0.5185 degrees.
To find the measure of the angle subtended by the diameter to the Moon as measured by an astronomer on the surface of Earth, we can use trigonometry.
First, let's draw a diagram to visualize the situation. Imagine the Earth, with an astronomer standing on its surface. From that point, draw a line to the center of the Earth and extend it to the Moon, which is directly overhead.
Now, we have a right triangle with one leg representing the radius of the Earth (approximately 6,371 km) and the hypotenuse representing the sum of the radius of the Earth and the orbital radius of the Moon (approximately 390,771 km).
Using trigonometry, we can use the tangent function to find the angle subtended by the diameter to the Moon:
tanθ = (opposite/adjacent)
In this case, the opposite side is the radius of the Earth and the adjacent side is the sum of the radius of the Earth and the orbital radius of the Moon.
tanθ = (6,371 km) / (390,771 km)
Now, let's calculate the angle in radians.
θ = arctan((6,371 km) / (390,771 km))
Using a scientific calculator or an online calculator, we can find that arctan(6,371 / 390,771) ≈ 0.0164 radians (rounded to four decimal places).
Finally, let's convert the angle from radians to degrees.
1 radian is equal to approximately 57.3 degrees.
θ ≈ 0.0164 radians * 57.3 degrees/radian
θ ≈ 0.9406 degrees (rounded to four decimal places)
Therefore, the measure of the angle subtended by the diameter to the Moon as measured by an astronomer on the surface of Earth is approximately 0.0164 radians or 0.9406 degrees.