The diameter of Earth is 12,742 kilometers; the diameter of the moon is 3476 kilometers.
a. If you flew around Earth by following the equator at a height of 10 kilometers, how many trips around the Moon could you take in the same amount of time, at the same height from the Moon and a t the same speed? Explain
b. The circle whose circumference is the equator is sometime call the ?great circle? of Earth or of the Moon. Use a ratio to compare the areas of the great circles of Earth and the Moon.
a.
trip around earth is pi*(12742+10) = 40061 km
trip around moon is pi * (2476+10) = 7810 km
so, what's 40061/7810?
b. ratio of great circles is the same as the ratio of the radii, since C = 2pi*r
a. To solve this problem, we need to calculate the length of the equator of Earth when flying at a height of 10 kilometers. Then we divide this length by the circumference of the Moon at the same height to find out how many trips around the Moon can be taken in the same amount of time.
1. Calculate the length of the equator of Earth at a height of 10 kilometers:
- The diameter of Earth is given as 12,742 kilometers.
- The radius of Earth can be found by dividing the diameter by 2: radius = diameter / 2 = 12,742 km / 2 = 6,371 km.
- The circumference of a circle can be calculated using the formula C = 2πr, where C is the circumference and r is the radius.
- Replace the values in the formula: C = 2π * 6,371 km ≈ 40,075 km (approximately).
2. Calculate the circumference of the Moon at a height of 10 kilometers:
- The diameter of the Moon is given as 3,476 kilometers.
- Following the same steps as above, we find that the circumference of the Moon at a height of 10 kilometers is approximately 21,788 km (approximately).
3. Divide the length of the equator of Earth by the circumference of the Moon:
- Number of trips = Length of Earth's equator / Circumference of Moon
- Number of trips ≈ 40,075 km / 21,788 km ≈ 1.84 trips (approximately).
Therefore, you could take approximately 1.84 trips around the Moon in the same amount of time, at the same height from the Moon, and at the same speed, if you flew around Earth by following the equator at a height of 10 kilometers.
b. To compare the areas of the great circles of Earth and the Moon, we can use the ratio of their circumscribed circles.
1. The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
2. The Earth's radius is 6,371 kilometers, so its area is A_Earth = π * (6,371 km)^2.
3. The Moon's radius is 1,738 kilometers (diameter of 3,476 km / 2), so its area is A_Moon = π * (1,738 km)^2.
4. Divide the area of Earth by the area of the Moon to find the ratio: Ratio = A_Earth / A_Moon.
5. Calculate the ratio: Ratio ≈ (π * (6,371 km)^2) / (π * (1,738 km)^2) ≈ (6,371 km)^2 / (1,738 km)^2 ≈ 14.59.
Therefore, the ratio of the areas of the great circles of Earth and the Moon is approximately 14.59.