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Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643

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BuyFindarrow_forward

Multivariable Calculus

8th Edition
James Stewart
ISBN: 9781305266643
Textbook Problem

The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 110 km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus.

To determine

To Find: The equation of this ellipse using the perilune altitude and apolune altitude.

Explanation

Given:

The perilune and apolune altitude are 110km and 314km . The radius of the moon is 1728km .

Calculation:

The center of the ellipse (h,k) is (0,0) .

Let the nearest surface of the moon (focus) be (rc) and the farthest from the surface of the moon (focus) be (r+c) .

Draw the ellipse formed by the moon as below.

Compute the value of a using the below equation.

2a=(r+c)+(rc)

Substitute the value of 1728 for r , 110 for c and 314 for c .

2a=(1728+314)+(1728(110))2a=2042+18382a=3880a=38802a=1940

Therefore, the value of a is 1940_ .

Compute the value of c from the below equation.

2c=(r+c)(rc)

The value of r that is the radius of the moon became 0 . The focus is at the center of the moon.

Substitute the value of 0 for r , 110 for c and 314 for c .

2c=(0+314)(0(110))2c=204c=2042c=102

Therefore, the value of c is 102_

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