Doubling Time

Date: 03/08/2002 at 14:50:39
From: Ariele McWhinney
Subject: Scientific math formula

In my science lesson this week they are talking about the doubling 
time of the human population and other things.  I have a formula that 
has DT standing for doubling time:
 
   DT = 70(or any other #...I used x)/% growth per unit time

I am very confused. I kind of understand the formula, but then they 
give me problems to figure out:

If the cost of an all-day ski lift ticket at Colorado's Vail Ski Area 
has been growing at about 7% per year ever since Vail opened in 1963, 
and at that time the cost of a ticket was $5, what did it cost in 
1993?

I have tried replacing the information that they give me in the 
formula, but I don't know which part of the formula to replace it 
with.

Ariele McWhinney

Date: 03/08/2002 at 16:06:05
From: Doctor Rick
Subject: Re: Scientific math formula

Hi, Ariele.

Here is a hint on finding things on Dr. Math. You can use the Search 
Dr. Math form

  http://mathforum.org/mathgrepform.html   

to search for words or a phrase relating to your question. For 
instance, if you type in

  doubling time

and select "that exact phrase" to look for these words together as a 
phrase, you'll find this answer:

  Rule 0f 72
  http://mathforum.org/dr.math/problems/vasseur1.26.99.html   

Here you will read about your formula, why it (sort of) works, and 
when it doesn't work. The number in the formula does matter, and it 
should be about 72; but the formula is only an approximation. The 
exact formula is

  DT = log(2)/log(1+p/100)

where p is the percentage increase per unit time. Thus, in your 
example, p=7, so 1+p/100 = 1.07 and

  DT = log(2)/log(1.07) = 10.245 years

However, this doesn't directly solve your problem, which says nothing 
about doubling time. Instead we can write this equation:

  C = 5(1.07)^t

where C is the cost of a lift ticket and t is the number of years 
since 1963. Do you understand this equation? At  t= 0 (the year 1963) 
the cost is $5:

  C = 5(1.07)^0
    = 5(1)

At t = 1 (the year 1964) the cost is 1.07 times $5; that's an increase 
of 7%. Each year following, you multiply by another factor of 1.07.

You can find the cost of a lift ticket in 1993 by deciding what the 
value of t is in that year, then plugging this value into the equation 
in place of t, and evaluating it to find C.

I'll be glad to answer any questions my explanation has raised. I 
don't know how much you understand, so I haven't tried to explain 
everything; I'll expect you to ask if you need more detail on some 
point.

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/   

Date: 03/08/2002 at 16:31:39
From: Ariele McWhinney
Subject: Scientific math formula

Thank you very much for your explanation; it will help me greatly. I 
think that my dad and I can do the rest.  I am in algebra this year 
and just getting into complicated formulas.

Thank you again,
Ariele McWhinney