Algebra 1 Word Problems - DRT
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It's a Bird, It's a Plane
Word Problems - DRT

 
 Now a bit of info about flying saucers (and airplanes too).
 The speed that an airplane travels over the ground
 is a combination of two things.
 
 One is the speed that it moves through the air 
 (called airspeed).
 The other is the speed that the air is moving
 (called wind).
 
 Human pilots try to fly routes where the wind is blowing
 in the same direction that they are flying.
 That lets them move faster over the ground.
 They get to the destination faster and use less fuel.
 (Alien's probably do the same thing,
 but I've never had the chance to ask.)
 
  Example:
 
 The wind is blowing from the north to the south
 at 10 miles per hour.
 
 Two airplanes leave the same airport at the same time
 AND TRAVEL AT THE SAME AIRSPEED.
 
 One travels north (into the wind)
 The other travels south (with the wind)
 
 At some time later, the one that was traveling north
 is 200 miles from the airport.
 The one traveling south is 235  miles from the airport.
 

 What is the airspeed of the two planes?

 
 The problems are starting to get more involved.
 We are going to use:
 
 D for distance
 R for rate (the speed the plane flies OVER THE GROUND)
 T for time.
 
 We also need to tell the two planes apart.
 To do this, we call the plane going north plane 1.
 Then we can call the distance it went D1
 and the speed it went over the ground R1
 and the time it traveled T1.
 
 We can also call the plane that went south, plane 2.
 Plane 2 can use D2 , R2 , and T2 for it's distance, rate and time.
 
 So what of these things do we know?
 
D1 = 200 miles D2 = 235 miles
R1 = airspeed - 10 R2 = airspeed + 10
T1 = unknown, but equal to T2 T2 = unknown, but equal to T1
 
 Here's what we do.
 We have too many things that are unknown.
 We need to get rid of a few of them.
 
 We don't know either of the times,
 but we know that they are the same.
 That means we can write them both as T.
 
 The ground speeds of the two planes
 are different because of the wind.
 But the airspeeds are the same.
 If we call the airspeed r, then
 R1 = r -10 and R2 = r + 10.
 
 The DRT equations for the two planes are:
 

D1 = R1 x T     and     D2 = R2 x T

 
 Substituting what we know, we have ...
 

200 = (r - 10) x T     and     235 = (r + 10) x T

 
 Now we have two equations with 2 unknowns.
 There are a lot of ways that we can solve these.
 
 In the way we are going to use,
 the first step is to turn them each around to be:
 

 T = "Stuff"

 

 
 Here's the tricky part.
 Since "T" is the same in both equations,
 we can say ...
 

 
 and actually, we don't even need the "T" ...
 

 
 Now we solve for "r" (which is the airspeed)
 
 Multiply both sides by (r - 10)(r + 10) to get rid of the denominators,
 and simplify ...
 

 
 So airspeed is about 124.29 miles per hour.
 Hmm, I don't think that these planes are F-16's.
 
 OK, now we look back at the original problem,
 and make sure we have actually answered 
 what they were looking for.
 
 We have.
 
 They MIGHT have asked for something else,
 like the Time they have been flying, 
 or the groundspeed of one or both planes.
 
 They didn't.
 

   copyright 2005 Bruce Kirkpatrick

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