Finding constants in exponential functions

[LivvyS]

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 modeled by the exponential growth function.

P(t) = Ae^kt

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

Homework Equations

P(t) = Ae^kt

The Attempt at a Solution

P(0) = 8.23x10⁶ so Ae^0k= 8.23x10⁶

P(10) = 9.77x10⁶ so Ae^10k= 9.77x10⁶Divide to eliminate A:
Ae^0k= 8.23x10⁶ / Ae^10k= 9.77x10⁶ = e^0k-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:
Ae^10k= 9.77x10⁶
Ae^{10*0.01917945693}= 9.77x10⁶
A = 9.77x10⁶ / e^{10*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 can't seem to produce 8.23x10⁶ at 0 years since 2000 and I can't seem to see why, any help would be much appreciated!

 

[Quantum Defect]

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 modeled by the exponential growth function.

P(t) = Ae^kt

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

Homework Equations

P(t) = Ae^kt

The Attempt at a Solution

P(0) = 8.23x10⁶ so Ae^0k= 8.23x10⁶

P(10) = 9.77x10⁶ so Ae^10k= 9.77x10⁶


Divide to eliminate A:
Ae^0k= 8.23x10⁶ / Ae^10k= 9.77x10⁶ = e^0k-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:
Ae^10k= 9.77x10⁶
Ae^{10*0.01917945693}= 9.77x10⁶
A = 9.77x10⁶ / e^{10*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 can't seem to produce 8.23x10⁶ at 0 years since 2000 and I can't 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?

 

[LivvyS]

Thanks for the reply!

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 what's wrong with it?

 

[BvU]

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

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

 

[Ray Vickson]

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 modeled by the exponential growth function.

P(t) = Ae^kt

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

Homework Equations

P(t) = Ae^kt

The Attempt at a Solution

P(0) = 8.23x10⁶ so Ae^0k= 8.23x10⁶

P(10) = 9.77x10⁶ so Ae^10k= 9.77x10⁶Divide to eliminate A:
Ae^0k= 8.23x10⁶ / Ae^10k= 9.77x10⁶ = e^0k-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:
Ae^10k= 9.77x10⁶
Ae^{10*0.01917945693}= 9.77x10⁶
A = 9.77x10⁶ / e^{10*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 can't seem to produce 8.23x10⁶ at 0 years since 2000 and I can't 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]

Poster had everything OK until he typed 997 instead of 977