# Thread: College Algebra Word Problem Help!

1. ## College Algebra Word Problem Help!

Questions:
In 1970, the population of the U.S. was 203.3 million. By 2003, it had grown to 294 million. Write the exponential growth function that models the data. Use the function to find the year the poulation will reach 315 million.

2010

I'm not for sure how to go about this problem. Help would be greatly appreciated.

2. Use the model $\displaystyle P=Ae^{kt}$

With the points $\displaystyle (t,P) = (1970,203.3),(2003,294)$

find k and A.

After this make $\displaystyle P=315$ and solve for t

3. Originally Posted by The Masked Trumpet
Questions:
In 1970, the population of the U.S. was 203.3 million. By 2003, it had grown to 294 million. Write the exponential growth function that models the data. Use the function to find the year the poulation will reach 315 million.

2010

I'm not for sure how to go about this problem. Help would be greatly appreciated.
Use the formula:
$\displaystyle A(t) = A_{0}e^{kt}$

Let 1970 be year "0". (The starting year)

$\displaystyle A(0) = A_{0} = 203.3 \ mil$

And we are given after 33 years (the year 2003):
$\displaystyle A(33) = A_{0}e^{k(33)} = 294 \ mil$

Using the above and solving for k we find:
$\displaystyle k = \frac{\ln \left( \frac{294}{203.3} \right)}{33}$

Now let's solve what we were asked.

We want $\displaystyle A(t) = 315 \ mil$

Use the formula once again: (But now we know the value for $\displaystyle k$, and we've had $\displaystyle A_{0}$ from the start)
$\displaystyle A(t) = A_{0}e^{kt}$

By just plugging in and solving for $\displaystyle t$:
$\displaystyle t = \frac{\ln \left( \frac{315}{203.3} \right)}{\left( \frac{\ln \left( \frac{294}{203.3} \right)}{33} \right)}$

We find:
$\displaystyle t = 39.171816$

I'm assuming your lecturer just rounded up to 40.

Then 1970 + 40 = 2010