Question

Having survived the meteor impact, thanks to some last minute evasive maneuvers, the Mathonauts now set their sights on their Interstellar Headquarters. The Interstellar Headquarters orbits the Earth based on the equation y2 + x2 = 40,000. Using the original trajectory of the ship and complete sentences, explain to the pilot how to find where the ship’s path will cross the Interstellar Headquarters’s path.

Answer 1

depends on your ships path.. which is whatever you set it to before..
use something simple like y=x+1

now if the orbit is x^2 + y^2 = 40000

then use substitution to find the x,y of your intersection.

x^2 + (x+1)^2 = 40000

First convert that into a quadratic.

IE: something that equates to 0.
then solve for x.. and you will have your x coordinate.

Now that you know your x coordinate you and given y =x+1
you can plug in your x coord and get your y.

y = (YOUR_x) + 1

You will now have your intercept for HQ

to finish it off.. plug the formula into geogebra and plot a pretty little graph, you'll score extra points that way.. (or at least you wont lose any - check the rubric.. the requirement is hidden in there)

Answer 2

If you're not sure how to convert that into a quadratic..
x^2 + y^2 = 40000

Replace y with the equality for y = (x+1)
x^2 + (x+1)^2 = 40000 (see how y was replaced with 'x+1'
x^2 + (x+1)(x+1) = 40000 FOIL The (X+1) part...
then you want to move the terms around until you have it in the quadratic form

ax^2 + bx + c = 0

Once you have the quadratic the value of x is given by the quadratic formula
\[x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{ 2a } \]

plug a,b,and c in, then solve for x and you are on your way.