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Two Hot-Air Balloons

Date: 10/21/96 at 18:4:26
From: www.
Subject: Solving Two Equations

Two hot-air balloons are flying over a field.  The height h, in feet, 
of the first balloon at time t seconds is h(t)=50+34t.  The height g, 
in feet, of the second balloon at time t is g(t)=400-16t.

a. In how many seconds will the balloons be at the same height?
b. How high will they be at that time?

The back of my book says that the answers are (a) 7 seconds and (b) 
288ft, but I can't do this myself:

   h(t) = 50+34t
    34t = -h(t)-50      all divided by 34
      t = -ht/34-1.47

   h(t) = 50+34t        all divided by t
      h = 50/t+34

   g(t) = 400-16t       all divided by t
      g = 400/t-16

   g(t) = 400-16t
    16t = 400-g(t)      all divided by 16
      t = 25-gt/16

What do I do next?

Date: 10/22/96 at 17:46:16
From: Doctor Allen
Subject: Re: Solving Two Equations

First - the calculations you have already performed.

h(t) = 50 + 34t

solving for 34t gives

34t = h(t)-50 [you had a -h(t)]

h(t) and g(t) are both written to express that h and g can be 
calculated from a value of t. The expression, h(t), is read as 
"h as a function of t".  Values of t can be used to find h, for 
example h when t=10 which is written as h(10)= 50 + 34 * 10 = 390. 
When you solved some of your equations, you read h(t) as 
h times t and cancelled the t from the expression.

  h(t) = 50 + 34t all divided by t
h(t)/t = 50/t + 34 [you had h(t)/t = h which is not correct]

The same is true for your work with the g(t) equation.

Second - The problem you describe involves finding the solutions 
to h, g, and t for a given point (when the balloons are at the same 
height). To find the answer, you most combine the two equations. 
When the balloons are at the same height, h(t) = g(t) since h and g 
are the heights of the two balloons. Starting from this point, we 
substitute for h(t) and g(t) and solve.

    h(t) = g(t)
50 + 34t = 400 - 16t    (substituting for h(t) and g(t))
     34t = 350 - 16 t   (subtract 50 from both sides)
     50t = 350          (add 16t to both sides)
       t = 7            (divide both sides by 50)

a) The balloons will reach the same height in 7 seconds

to find h(t) and g(t), substitute 7 for t in both equations

h(t) = 50 + 34t
h(7) = 50 + 34*7 = 288

Since the balloons are supposed to be at the same height, g should be 
288 as well. Let's check.

g(t) = 400 - 16t
g(7) = 400 - 16*7 = 288

b) The balloons will be 288 feet up.

-Doctor Allen,  The Math Forum
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Associated Topics:
High School Basic Algebra

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