SOLUTION: 3. Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: 3. Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.       Log On


   

Question 155378: 3. Steve traveled 200 miles at a certain speed. Had he gone 10mph faster, the trip would have taken 1 hour less. Find the speed of his vehicle.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
# 3

Let x = original speed (ie slower speed)


d=rt Start with the distance-rate-time formula


200=%28x%29t Plug in d=200 and r=x


200%2Fx=t Divide both sides by "x" to isolate "t"


The statement "if he had gone 10mph faster, the trip would have taken 1 hour less" tells us that the new speed is x%2B10 and the new time is t-1


d=rt Go back to the distance-rate-time formula


200=%28x%2B10%29%28t-1%29 Plug in d=200, r=x%2B10, and replace t with t-1


200=xt-x%2B10t-10 FOIL


200=x%28200%2Fx%29-x%2B10%28200%2Fx%29-10 Plug in t=200%2Fx


200=200-x%2B2000%2Fx-10 Multiply


200x=200x-x%5E2%2B2000-10x Multiply every term by the LCD "x" to clear the denominator


0=200x-x%5E2%2B2000-10x-200x Subtract 200x from both sides


0=-x%5E2-10x%2B2000 Combine like terms


Notice we have a quadratic equation in the form of ax%5E2%2Bbx%2Bc where a=-1, b=-10, and c=2000


Let's use the quadratic formula to solve for x


x+=+%28-b+%2B-+sqrt%28+b%5E2-4ac+%29%29%2F%282a%29 Start with the quadratic formula


x+=+%28-%28-10%29+%2B-+sqrt%28+%28-10%29%5E2-4%28-1%29%282000%29+%29%29%2F%282%28-1%29%29 Plug in a=-1, b=-10, and c=2000


x+=+%2810+%2B-+sqrt%28+%28-10%29%5E2-4%28-1%29%282000%29+%29%29%2F%282%28-1%29%29 Negate -10 to get 10.


x+=+%2810+%2B-+sqrt%28+100-4%28-1%29%282000%29+%29%29%2F%282%28-1%29%29 Square -10 to get 100.


x+=+%2810+%2B-+sqrt%28+100--8000+%29%29%2F%282%28-1%29%29 Multiply 4%28-1%29%282000%29 to get -8000


x+=+%2810+%2B-+sqrt%28+100%2B8000+%29%29%2F%282%28-1%29%29 Rewrite sqrt%28100--8000%29 as sqrt%28100%2B8000%29


x+=+%2810+%2B-+sqrt%28+8100+%29%29%2F%282%28-1%29%29 Add 100 to 8000 to get 8100


x+=+%2810+%2B-+sqrt%28+8100+%29%29%2F%28-2%29 Multiply 2 and -1 to get -2.


x+=+%2810+%2B-+90%29%2F%28-2%29 Take the square root of 8100 to get 90.


x+=+%2810+%2B+90%29%2F%28-2%29 or x+=+%2810+-+90%29%2F%28-2%29 Break up the expression.


x+=+%28100%29%2F%28-2%29 or x+=++%28-80%29%2F%28-2%29 Combine like terms.


x+=+-50 or x+=+40 Simplify.


So the possible answers are x+=+-50 or x+=+40

Since a negative speed doesn't make sense, this means that the only solution is x=40

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Answer:

So the original speed was 40 mph