A space camera circles the Earth at a height of h miles above the surface. Suppose that d distance, IN MILES, on the surface of the Earth can be seen from the camera.
(a) Find an equation that relates d and h.
(b) If d is to be 3500 miles, how high must the camera orbit above Earth?
(c) If the camera orbits at a height of 400 miles, what distance d on the surface can be seen?
surely you could have looked up the radius of the earth for us, which is needed to do the question.
A space camera circles the Earth at a height of h miles above the surface. Suppose that d distance, IN MILES, on the surface of the Earth can be seen from the camera.
(a) Find an equation that relates d and h.
(b) If d is to be 3500 miles, how high must the camera orbit above Earth?
(c) If the camera orbits at a height of 400 miles, what distance d on the surface can be seen?
Draw a circle with center O. Locate a point C outside the circle.
Draw lines from point C to the tangency points on both sides of the circle.
The arc length between these two tangency points is the distance "d" visible to the camera.
With the earth's radius of 3963 miles. the central angle between the two tangency points is defined by µ = (2)arccos(3963/h + 3963))º.
Therefore, the visible distance "d" = Rµ, where µ is in radians.
This should enable you to find the other two answers you seek.
Here is the raidus of Earth as given in the book:
3960 miles.
Here is the raidus of Earth as given in the book:
3960 miles.
(a) To find an equation that relates d and h, we can use the concept of similar triangles.
Consider a right triangle formed by the radius of the Earth (R), the height of the camera above the surface (h), and the distance on the surface that can be seen (d).
The length of the hypotenuse of the triangle is the radius of the Earth plus the height of the camera, which is R + h. The length of the base of the triangle is just the radius of the Earth, R.
Since the triangles are similar, we can set up the following proportion:
(R + h) / R = d / R
Cross-multiplying gives:
(R + h) = d
This equation relates d and h.
(b) If we want to find the height of the camera (h) when the distance on the surface (d) is 3500 miles, we can use the equation derived in part (a).
Substituting d = 3500 into the equation:
(R + h) = 3500
Since h is the height above the surface, we can solve for h:
h = 3500 - R
The height of the camera must be 3500 minus the radius of the Earth.
(c) If the camera orbits at a height of 400 miles, we can again use the equation derived in part (a) to find the distance on the surface that can be seen (d).
Substituting h = 400 into the equation:
(R + 400) = d
The distance on the surface that can be seen is given by the sum of the radius of the Earth and the height of the camera. So, in this case, the distance d is equal to R + 400.