Asma A.

asked • 03/11/14# Doubling time

## 3 Answers By Expert Tutors

Steve S. answered • 03/12/14

Tutoring in Precalculus, Trig, and Differential Calculus

Since populations grow continuously, we should use a continuous exponential growth model. In fact, this is the model defined as Exponential Growth (see http://mathworld.wolfram.com/ExponentialGrowth.html)

A = P e^(r t)

If the initial population is 1 unit, then in 15 years the population will be 2 units.

2 = 1 e^(15 r)

Take the Natural Logarithm of both sides:

ln(2) = 15 r

r = ln(2)/15 per year, exactly, or

r ≈ 4.620981203733% per year, approximately (no matter how many decimals you get from a calculator the answer is always approximate).

Since no mention was made of how to round, I give both the exact answer and the most decimals I can get out of my calculator for an approximate answer.

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To use the Compound Interest Formula instead of the Exponential Growth Formula in this problem is wrong. Here’s why:

The Compound Interest Formula states A = P(1 + r/n)^(nt) where the interest is calculated ONLY at integer multiples of t/n; i.e., every 1/n time units (e.g., years).

So A(t) is actually the step function,

A(t) = P(1 + r/n)^(floor(nt)).

As a visualization example, this GeoGebra graph:

http://www.wyzant.com/resources/files/264949/continuous_exponential_vs_discrete_compound_growth

shows:

S(t) = (1 + 0.05/2)^(floor(2 t)),

C(t) = e^(0.05 t), and their difference

D(t) = C(t) – S(t).

You can clearly see that for t > 0, D > 0 and increasing on average.

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Why does this level of detail matter, anyway?

Well, if a bridge an engineer (you?) designed collapses due to a design flaw, “I’m sorry!” will not be a strong defense in the ensuing multibillion dollar law suits brought by the survivors of the victims.

Parviz F.

^{k}= P

^{n}

03/12/14

Steve S.

A = P(1 + r)^t

2 = 1(1 + r)^15

2^(1/15) = 1 + r

r = 2^(1/15) - 1 ≈ 4.729412282063 %

Notice that this growth rate is larger than the one calculated for the Continuous Exponential growth rate above. AND A(t) is a step function that holds its value for a year after every interest calculation and deposit into the account.

03/12/14

Parviz F. answered • 03/12/14

Mathematics professor at Community Colleges

Steve S.

03/12/14

Parviz F.

^{n}= e

^{x}

03/12/14

Kay G. answered • 03/12/14

~20 Years Accounting Tutoring Experience

^{n}

^{15}where you are solving for r.

^{20}

^{20}

^{20}]

Steve S.

03/12/14

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Kay G.

yourtextbook, or if it agrees with Steve.03/13/14