The distance, "d", to the horizon (= farthest point on the earth still visible) for an object that is "h" miles above the surface of the earth is given the formula d = the square root of 8000h +h^2. Determine how many miles above the surface of the earth is an orbiting satellite if the distance to the horizon is 900 miles.
just plug in your numbers and solve for h
900 = √(h^2+8000h)
h^2+8000h-900^2 = 0
h = 100
To determine how many miles above the surface of the earth an orbiting satellite is, we need to solve the equation d = √(8000h + h^2) for h, where d is the given distance to the horizon. In this case, the distance to the horizon is 900 miles.
Let's substitute the given values into the equation:
900 = √(8000h + h^2)
To solve for h, we need to isolate h on one side of the equation. To do this, we will square both sides of the equation:
(900)^2 = (√(8000h + h^2))^2
Simplifying both sides of the equation:
810,000 = 8000h + h^2
Rearranging the equation in standard form:
h^2 + 8000h - 810,000 = 0
This is a quadratic equation, and we can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
Applying this formula to our equation:
For the equation h^2 + 8000h - 810,000 = 0, a = 1, b = 8000, and c = -810,000.
h = (-8000 ± √(8000^2 - 4(1)(-810,000))) / (2*1)
Simplifying further:
h = (-8000 ± √(64,000 ,000 + 32,400 ,000)) / 2
h = (-8000 ± √(96,400 ,000)) / 2
h = (-8000 ± 9,819.62) / 2
Now, we have two possible values for h:
h1 = (-8000 + 9,819.62) / 2 = 409.81 miles
h2 = (-8000 - 9,819.62) / 2 = -8,819.62 miles
Since distance above the Earth's surface cannot be negative, the satellite must be approximately 409.81 miles above the surface of the Earth.