Expert: Paul Klarreich - 12/4/2006

Question
I am working on a problem that states -you are in a hot air balloon and the distance you can see depends on how high you are, write the function with respect to the following information -
Height (ft)   15      30      45      60      75     90
Distance (km) 13.883  19.562  23.959  27.665  30.931 33.883

I am totally stuck.  Please help!  Thank you, Keely

Answer
Questioner:  Keely
Category:  Advanced Math
 
Subject:  Algebra II exponential functions
Question:  I am working on a problem that states -you are in a hot air balloon and the distance you can see depends on how high you are, write the function with respect to the following information -

Height (ft)     15      30      45      60      75     90
Distance (km) 13.883  19.562  23.959  27.665  30.931 33.883

I am totally stuck.  Please help!  Thank you, Keely
 
WARNING: THE FOLLOWING DISCUSSION MAY CONTAIN FRACTIONS AND OTHER MATERIAL INAPPROPRIATE FOR CERTAIN COMPUTING SYSTEMS.  BE SURE TO VIEW IT IN A FIXED-SIZE FONT, SUCH AS COURIER.
............................
Hi, Keely,

Since you did say something about exponential functions, this suggests looking at the ratios of terms.  For example:

30 = 2(15) and the ratio of their values is 1.409061442
60 = 2(30) and the ratio of their values is 1.41422145
90 - 2(45) and the ratio of their values is 1.414207605

Hmmmm.. those are just about sqrt(2).  How about:

45 = 3(15) and the ratio of their values is 1.725779731
90 = 3(30) and the ratio of their values is 1.732082609

and those are just about sqrt(3).  So it looks as if:

f(x2)   
----- = sqrt(x2/x1)
f(x1)

Now suppose we just let x2 = x and  x1 = 15.  Then:

f(x)
----- = sqrt(x/15)
f(15)

Since f(15) is 13.883, we can write:

f(x) = 13.883 sqrt(x/15)

and we have a function.