Question

3. 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 y^2 + x^2 = 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

@woohoo

Answer 2

@phi

Answer 3

i need serious help!

Answer 4

y^2 + x^2 = 40,000
is the equation of a circle, centered at (0,0) , with radius = 200 (notice 200*200= 40000)

they say
Using the original trajectory of the ship

I assume you have an equation ? we need it to answer this question

Answer 5

my equation is y=5x+1 @phi

Answer 6

the problem is asking where your line intersects the circle.

Answer 7

Im still lost i don`t get it at all @phi

Answer 8

Here is a graph of the problem

Answer 9

you have two equations
y=5x+1
y^2 + x^2 = 40,000

notice that if you replace y in the second equation using y= 5x+1 you get
\[ (5x+1)^2 +x^2 = 40000 \]
first expand (5x+1)(5x+1)
can you do that ?

Answer 10

which means i have to use foil? @phi

Answer 11

yes

Answer 12

so i got 25^2 + 10x + 5x + 1 ?

Answer 13

@phi

Answer 14

F(irst) 5x*5x or 25x^2
O(uter) 5x*1 or 5x
I(nner) 1*5x or 5x
L(ast) 1*1 or 1

if you combine the 5x+5x to 10x we get
25x^2 +10x+1

so you are close

Answer 15

you now have
\[ (5x+1)^2 +x^2 = 40000 \\ (5x+1)(5x+1) +x^2 = 40000 \\
25x^2 +10x+1 + x^2 = 40000 \]

now combine the "like terms". In this case the x^2 terms.
also add -40000 to both sides.
can you do that ?

Answer 16

so with the like terms with x^2 and 25x^2 which give you 26x^2
@phi

Answer 17

yes. You can think: 25 x squarers plus one more x squared is 26 x squarers
(I think of x^2 as a thing we count, and we have 26 of them)

\[ 25x^2 +10x+1 + x^2 = 40000 \\26x^2 +10x+1 = 40000
\]

now add -40000 to both sides.

Answer 18

to add -40000 to both sides you write -40000 on both sides

Answer 19

both sides? @phi

Answer 20

im not sure if i did it right because i subtract 40000 from 10x and got 1,5.38

Answer 21

yes. If you have an equation, it will stay an equation if you add the same thing to both sides. (to keep things fair, so to speak)

Answer 22

when you add letters and numbers, you don't get *just* numbers.

Here, to add -40000 to both sides we do this
\[ 26x^2 +10x+1-40000 = 40000 -40000 \]

Answer 23

notice we added -40000 to both sides.

now simplify. and you can only combine *like terms*
so the x terms don't get involved.

Answer 24

im confused @phi

Answer 25

\[ 26x^2 +10x+1-40000 = 40000 -40000 \]
what is 1-40000 ? on the left side ?

Answer 26

39,999 ? @phi

Answer 27

the correct numbers, but the answer is negative: -39999

Answer 28

what about the right hand side of the = sign ? what is 40000 - 40000
?

Answer 29

0? @phi

Answer 30

yes. so now you have
\[ 26x^2+10x-39999=0 \]

that is a "quadratic equation" (because it has one variable , the x, and the highest exponent on the x is a 2

to solve for x, you use the quadratic formula. It is more complicated than the algebra we have done so far, so you probably should watch Khan's video (maybe a few times!) and then try using it. You will need a calculator. You should get the same numbers as in the graph I posted up above.
http://www.khanacademy.org/math/algebra/quadratics/quadratic_formula/v/using-the-quadratic-formula