When they meet, how far are they from Denver?

Question

An airplane leaves Los Angeles for Denver at a speed of 420 miles per hour. Thirty minutes later, a plane going from Denver to Los Angeles leaves Denver, which is 850 miles from Los Angeles, at a speed of 520 miles per hour. When they meet, how far are they from Denver?

Answer 1

You need to use a bit of algebra to solve this problem, but for a problem like this, it is always good to write out what you know.

 

1. The total distance between the two cities is 850 miles

2. Plane one is traveling in the positive x-direction at 420 miles in one hour.

3. Plane two is traveling in the negative x-direction at 520 miles in one hour.

4. The second plane starts flying a half hour after the first

 

Okay, now its time to set up some equations.  The equation used here is d=rt, or Δx=vt.  These two equations mean the exact same thing.  For a question at this level, we will use the first equation, d=rt, or distance equals rate multiplied by time.

 

plane one: D=420t

plane two: D=520t

 

The distance traveled by plane one before plane two takes off is equal to (1/2)(420) = 210 miles; plane one took off a half hour earlier and it travels 420 miles in one hour.  Therefore, in a half hour, it travels 210 miles.

 

To find the distance between the planes when the second plane takes off simply subtract 210 from 850 to get 640 miles. This means that the planes will meet when the distance traveled by both planes adds together to be 640 miles.

 

Now its time to find time.  In order to do so, we create the equation, 420t + 520t = 640, or 940t = 640.  We can combine the two equations because we know that the planes will meet after a total distance of 640 miles is traveled. t=0.6809 hours.

 

For the first plane, which is traveling towards Denver, D = 210+(420)(0.6809) ~ 496 miles

For the second plane, which is traveling away from Denver, D = (-520)(0.6809) ~ -354 miles

 

The question asks how far the planes will be from Denver when they meet.  From our equations, we know the distances that both planes travel.  We also know that the second plane is traveling away from Denver, So, when the two planes meet in air, they will be 354 miles away from Denver. One can also use the first equation to find the correct answer.  The first equation involves the first plane which is traveling towards Denver from LA.  We know that the total distance between the two cities is 850 miles.  So, 850 - 496 = 354 miles.  

 

As you can see, one can find the correct answer using either equations.  There are also more difficult ways to approach this problem that I won't get into.  Problems like these are rather common in algebra one.  When I took the class, I always hated problems like these.  Nowadays, a problem like this comes real easy to me.  If you, or anyone else for that matter, has any more math or science related questions, please feel free to e-mail me at [email protected].  I won't always be available, and I'm not going to do your homework for you, but I will be able to guide you in the right direction if you are struggling.  

 

 

 

Answer 2

Let X be the distance from Los Angelas for the first airplane

Let t be the time from the start of the plane leaving Los Angelas

Let Y be the distance from Denver for the second pland

so

X = 420 Miles/hour * t

Y = 520 Miles/hour * (t-.5)  (The plane left a half hour later)

X + Y = 850 when the cross

Need to Solve for Y

X = 850 - Y and putting that into the first equation.

850 - Y = 420 * t

t = (850 - Y) / 420 

Putting t into the second equation

Y = 520 * (850 - Y)/ 420 -260

Y = 1052.38 - 1.238 Y -260

2.238 Y = 792.38

Y = 354.06 Miles

X =  495.94 Miles

t = 1.18 hours

Checking your work

X = 420 * 1.18 = 495.94

Y = 520 * ( 1.18 - .5) = 354

to X + Y = 850 Miles   (with rounding issues.)