SOLUTION: I need help with this word problem: A plane flies 720 miles against a steady 30mi/h head wind and then returns to the same point with the wind. If the entire trip takes 10h, wha

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I need help with this word problem: A plane flies 720 miles against a steady 30mi/h head wind and then returns to the same point with the wind. If the entire trip takes 10h, wha      Log On


   

Question 88835: I need help with this word problem:
A plane flies 720 miles against a steady 30mi/h head wind and then returns to the same point with the wind. If the entire trip takes 10h, what is the planes speed in still air?
Thank you for the help
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
d=rt Start with the blank rate equation


720=%28x%2B30%29%2At%5B1%5D Now plug in the given values. This is the rate equation with the wind.

720%2F%28x%2B30%29=t%5B1%5D Solve for t



720=%28x-30%29%2At%5B2%5D Now plug in the given values. This is the rate equation against the wind.

720%2F%28x-30%29=t%5B2%5D Solve for t

Now combine the two "t" equations and set them equal to 10

t%5B1%5D%2Bt%5B2%5D=10

720%2F%28x%2B30%29%2B720%2F%28x-30%29=10 Plug in t%5B1%5D=720%2F%28x%2B30%29 and t%5B2%5D=720%2F%28x-30%29


%28x%2B30%29%28x-30%29%28720%2F%28x%2B30%29%2B720%2F%28x-30%29%29=10%28x%2B30%29%28x-30%29 Multiply both sides by %28x%2B30%29%28x-30%29


720%28x-30%29%2B720%28x%2B30%29=10%28x%2B30%29%28x-30%29 Distribute the left side


720%28x-30%29%2B720%28x%2B30%29=10%28x%5E2-900%29 Foil the left side

720%28x-30%29%2B720%28x%2B30%29=10x%5E2-9000 Multiply the right side

720x-21600%2B720%28x%2B30%29=10x%5E2-9000 Distribute and multiply 720%28x-30%29

720x-21600%2B720x%2B21600=10x%5E2-9000 Distribute and multiply 720%28x%2B30%29

1440x=10x%5E2-9000 Combine like terms on the left side

0=10x%5E2-1440x-9000 Subtract 1440x from both sides


Now let's use the quadratic formula to solve for x:


Starting with the general quadratic

ax%5E2%2Bbx%2Bc=0

the general solution using the quadratic equation is:

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29

So lets solve 10%2Ax%5E2-1440%2Ax-9000=0 ( notice a=10, b=-1440, and c=-9000)

x+=+%28--1440+%2B-+sqrt%28+%28-1440%29%5E2-4%2A10%2A-9000+%29%29%2F%282%2A10%29 Plug in a=10, b=-1440, and c=-9000



x+=+%281440+%2B-+sqrt%28+%28-1440%29%5E2-4%2A10%2A-9000+%29%29%2F%282%2A10%29 Negate -1440 to get 1440



x+=+%281440+%2B-+sqrt%28+2073600-4%2A10%2A-9000+%29%29%2F%282%2A10%29 Square -1440 to get 2073600



x+=+%281440+%2B-+sqrt%28+2073600%2B360000+%29%29%2F%282%2A10%29 Multiply -4%2A-9000%2A10 to get 360000



x+=+%281440+%2B-+sqrt%28+2433600+%29%29%2F%282%2A10%29 Combine like terms in the radicand (everything under the square root)



x+=+%281440+%2B-+1560%29%2F%282%2A10%29 Simplify the square root



x+=+%281440+%2B-+1560%29%2F20 Multiply 2 and 10 to get 20

So now the expression breaks down into two parts

x+=+%281440+%2B+1560%29%2F20 or x+=+%281440+-+1560%29%2F20

Lets look at the first part:

x=%281440+%2B+1560%29%2F20

x=3000%2F20 Add the terms in the numerator
x=150 Divide

So one answer is
x=150



Now lets look at the second part:

x=%281440+-+1560%29%2F20

x=-120%2F20 Subtract the terms in the numerator
x=-6 Divide

So another answer is
x=-6

So our solutions are:
x=150 or x=-6




Since a negative speed doesn't make any sense, our solution is x=150

So the plane's speed in still air is 150 mph