# Finding constants in exponential functions

## Homework Statement

In 2000 the population of a country was estimated to be 8.23 million. In 2010 the population was 9.77 million.

Assume that the number of people P(t) in millions at time t (in years since 2000) is modelled by the exponential growth function.

P(t) = Aekt

Find P(t) giving the two constants in it to 2 significant figures.

P(t) = Aekt

## The Attempt at a Solution

P(0) = 8.23x106 so Ae0k= 8.23x106

P(10) = 9.77x106 so Ae10k= 9.77x106

Divide to eliminate A:
Ae0k= 8.23x106 / Ae10k= 9.77x106 = e0k-10k= 8.23x106 / 9.77x106 = e-10 k = 8.23x106 / 9.77x106

(I am not certain that this step is right)

-10k = ln 8.23x106 / 9.77x106

k = 0.01917945693

To find A:
Ae10k= 9.77x106
Ae10*0.01917945693= 9.77x106
A = 9.77x106 / e10*0.01917945693 = 8064904.714

These values for k and A seem to produce 9.77x106 when 10 (years) is plugged into the function, however I cant seem to produce 8.23x106 at 0 years since 2000 and I cant seem to see why, any help would be much appreciated!

Quantum Defect
Homework Helper
Gold Member

## Homework Statement

In 2000 the population of a country was estimated to be 8.23 million. In 2010 the population was 9.77 million.

Assume that the number of people P(t) in millions at time t (in years since 2000) is modelled by the exponential growth function.

P(t) = Aekt

Find P(t) giving the two constants in it to 2 significant figures.

P(t) = Aekt

## The Attempt at a Solution

P(0) = 8.23x106 so Ae0k= 8.23x106

P(10) = 9.77x106 so Ae10k= 9.77x106

Divide to eliminate A:
Ae0k= 8.23x106 / Ae10k= 9.77x106 = e0k-10k= 8.23x106 / 9.77x106 = e-10 k = 8.23x106 / 9.77x106

(I am not certain that this step is right)

-10k = ln 8.23x106 / 9.77x106

k = 0.01917945693

To find A:
Ae10k= 9.77x106
Ae10*0.01917945693= 9.77x106
A = 9.77x106 / e10*0.01917945693 = 8064904.714

These values for k and A seem to produce 9.77x106 when 10 (years) is plugged into the function, however I cant seem to produce 8.23x106 at 0 years since 2000 and I cant seem to see why, any help would be much appreciated!

The trick is to remember what the value of the exponential is when t = 0. What is exp(0)? If you know what the population is at t=0, how is this related to A?

So if the exponent of e is 0 then P(0) = A*1 so A = 8.23x106
But if this were true than the answer for a I calculated cannot be right, if so could you give me a clue as to whats wrong with it?

BvU
Homework Helper
Method is right, something goes wrong from -10k = ln 8.23x106 / 9.77x106 ⇒k = 0.01917945693

(you typed ... instead of ...)

Ray Vickson
Homework Helper
Dearly Missed

## Homework Statement

In 2000 the population of a country was estimated to be 8.23 million. In 2010 the population was 9.77 million.

Assume that the number of people P(t) in millions at time t (in years since 2000) is modelled by the exponential growth function.

P(t) = Aekt

Find P(t) giving the two constants in it to 2 significant figures.

P(t) = Aekt

## The Attempt at a Solution

P(0) = 8.23x106 so Ae0k= 8.23x106

P(10) = 9.77x106 so Ae10k= 9.77x106

Divide to eliminate A:
Ae0k= 8.23x106 / Ae10k= 9.77x106 = e0k-10k= 8.23x106 / 9.77x106 = e-10 k = 8.23x106 / 9.77x106

(I am not certain that this step is right)

-10k = ln 8.23x106 / 9.77x106

k = 0.01917945693

To find A:
Ae10k= 9.77x106
Ae10*0.01917945693= 9.77x106
A = 9.77x106 / e10*0.01917945693 = 8064904.714

These values for k and A seem to produce 9.77x106 when 10 (years) is plugged into the function, however I cant seem to produce 8.23x106 at 0 years since 2000 and I cant seem to see why, any help would be much appreciated!

It is much, much easier just to work in units of millions, so that your two equations are
$$A = 8.23 \; \text{and}\; A e^{10 k} = 9.77$$
By choosing those new units you eliminate a lot of the clutter and make it much easier to see what is happening (and to find errors). You can always convert to other units later, after you have solved the problem.

BvU