A satellite 200 miles above the earth is orbiting the earth once every 6 hours. How long, in hours, does it take the satellite to travel 8,400 miles? (Assume the radius of the earth is 4,000 miles.) Give your answer both as an exact value and an approximation to three significant digits.
Work out please
.1570
To find the time it takes for the satellite to travel 8,400 miles, we need to determine the circumference of the satellite's orbit.
The circumference of a circle is given by the formula: C = 2πr, where C is the circumference and r is the radius.
Given that the satellite is orbiting at a distance of 200 miles above the earth's surface, we can calculate the radius of the satellite's orbit as follows:
radius = radius of the earth + height of the satellite
radius = 4,000 miles + 200 miles
radius = 4,200 miles
Using the formula for circumference, we have:
C = 2πr
C = 2π(4,200)
C ≈ 26,389.38 miles
Since the satellite completes one orbit in 6 hours, we can determine the speed of the satellite by dividing the circumference by the time taken:
speed = circumference / time
speed = 26,389.38 miles / 6 hours
speed ≈ 4,398.23 miles/hour
To find the time it takes for the satellite to travel 8,400 miles, we divide the distance by the speed:
time = distance / speed
time = 8,400 miles / 4,398.23 miles/hour
time ≈ 1.91 hours
Therefore, it takes approximately 1.91 hours for the satellite to travel 8,400 miles.
As an exact value, the time can be expressed as 210/11 hours, which is approximately 1.91 hours (rounded to three significant digits).
To find the time it takes for the satellite to travel 8,400 miles, we can use the formula for the circumference of a circle. The distance traveled by the satellite in one orbit is equal to the circumference of the orbit, which can be calculated using the formula C = 2πr, where r is the radius of the orbit.
Given that the satellite is 200 miles above the earth and the radius of the earth is 4,000 miles, the radius of the orbit is the sum of the altitude and the radius of the earth. So, the radius of the orbit is 4,200 miles (4,000 + 200).
Using the formula for the circumference, we can find the distance traveled by the satellite in one orbit:
C = 2π * 4,200 miles
C ≈ 26,388.82 miles.
Since the satellite orbits the earth once every 6 hours, the time it takes for one orbit is 6 hours.
Now, we can set up a proportion to find the time it takes for the satellite to travel 8,400 miles.
Time / Distance = Time for one orbit / Distance for one orbit
Let x represent the time it takes for the satellite to travel 8,400 miles:
x / 8,400 miles = 6 hours / 26,388.82 miles
To solve for x, we can cross-multiply and divide:
x = (8,400 miles * 6 hours) / 26,388.82 miles
Calculating this expression, we get:
x ≈ 2.02 hours (approximation to three significant digits)
Therefore, it takes approximately 2.02 hours for the satellite to travel 8,400 miles.