Question

The first artificial satellite to orbit Earth was Sputnik I, launched by the Soviet Union in 1957. The orbit was an ellipse with Earth's center as one focus. The orbit's highest point above Earth's surface was 583 miles, and its lowest point was 132 miles.a) find an equation of the orbit.b) how far from Earth is the other focus?c) What is the length of the major axis?

Answer 1

a) The equation of the orbit is \(\frac{x^2}{(583+3963)^2}+\frac{y^2}{132^2} = 1\). b) The distance from the center to the other focus is calculated using the formula \(c = \sqrt{(583+3963)^2 - 132^2}\). c) The length of the major axis is \(2(583+3963)\).

Step-by-step explanation:

a) To find an equation of the orbit, we can use the standard form equation for an ellipse:
\[\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2} = 1\] where \((h,k)\) represents the center of the ellipse, \(a\) represents the distance from the center to the vertices along the major axis, and \(b\) represents the distance from the center to the co-vertices along the minor axis. In this case, the center of the ellipse is the Earth's center, which is at the origin \((0,0)\). The highest point of the orbit is 583 miles above the Earth's surface, so the distance from the center to the vertices along the major axis is 583 miles + the radius of the Earth. The radius of the Earth is approximately 3963 miles. The lowest point of the orbit is 132 miles above the Earth's surface, so the distance from the center to the co-vertices along the minor axis is 132 miles. Substituting the values into the equation:
\[\frac{x^2}{(583+3963)^2}+\frac{y^2}{132^2} = 1\]

b) The distance from the center to the other focus can be found using the relationship \(c = \sqrt{a^2 - b^2}\), where \(c\) represents the distance from the center to the foci. Substituting the values into the equation:
\[c = \sqrt{(583+3963)^2 - 132^2}\]

c) The length of the major axis can be found using the formula \(2a\), where \(a\) represents the distance from the center to the vertices along the major axis. Substituting the values into the equation:
\[2a = 2(583+3963)\]

Answer 2

A) (x²/127806.25) + (y²/76956) = 1

B) 451 miles

C) 715 miles

Step-by-step explanation:

Center is at (0,0) and focus is at (c,0) while vertex is at (a,0)

Thus, a - c = 132 miles

and a + c=583 miles

Adding both equations together, we have;

2a= 715 miles

Thus, a = 715/2

a = 357.5 miles

Since a+c=583

Thus, c = 583-357.5 = 225.5 miles

Now, for an ellipse it is defined by; a² = b² + c²

a² = 357.5² = 127806.25

c² = 225.5² = 50850.25

So, b² = 127806.25 - 50850.25

b² = 76956

b = √76956

b = 277.409

A) Equation of tangent to ellipse which is equation of orbit would be;

(x²/a²) + (y²/b²) = 1

Thus,

(x²/127806.25) + (y²/76956) = 1

B) distance between the earth and the other focus is = 2c = 2 x 225.5 = 451 miles

C) Length of other major axis = 2a = 2 x 357.5 = 715 miles