SOLUTION: The sales of a new electronic gadget are growing exponentially such that a total of 10000 gadgets were sold in 2000 and 82700 gadgets were sold in 2002. If this trend continues, in

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: The sales of a new electronic gadget are growing exponentially such that a total of 10000 gadgets were sold in 2000 and 82700 gadgets were sold in 2002. If this trend continues, in      Log On


   

Question 637300: The sales of a new electronic gadget are growing exponentially such that a total of 10000 gadgets were sold in 2000 and 82700 gadgets were sold in 2002. If this trend continues, in what year will sales of the gadget reach/exceed 500000?

I am unsure if I should be using the exponential growth function A=A(0)e^kt or not and if so how do I work out the value of k? Very confused. Thank you.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
You're not very confused because everything you figured out is correct. The only thing you're missing is how to find the "k".

To find the "k", you use the equation, A+=+A%5B0%5D%2Ae%5E%28kt%29, and the information you were given about the sales in 2002. In the equation...
A = the sales t years after the starting point, 2000
A%5B0%5D is the sales at the starting point, 2000. IOW, A%5B0%5D is the amount at t = 0.
e = Euler's constant
k = some constant
t = a number of years after the starting point
With the information you've been given, our
A%5B0%5D+=+10000
For the year 2002
A = 82700
t = 2 (Since 2002 is two years after 2000)
This makes our working equation:
82700+=+10000%2Ae%5E%28k%2A2%29
We can solve this for k. Dividing both sides by 10000:
8.27+=+e%5E%28k%2A2%29
Find the log of each side, using ln because of the "e":
ln%288.27%29+=+ln%28e%5E%28k%2A2%29%29
Using a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 we can move the exponent out in front:
ln%288.27%29+=+%28k%2A2%29%2Aln%28e%29
Since ln(e) = 1 this simplifies to:
ln%288.27%29+=+%28k%2A2%29
Dividing by 2 we get:
ln%288.27%29%2F2+=+k
This is an exact expression for k. We probably want to use a decimal so from our calculator we should get:
1.05631725 = k

Now that we have the k, we have a full equation:
A+=+10000%2Ae%5E%281.05631725t%29
which can then use this equation to answer other questions.

To find when the sales hit 500000:
500000+=+10000%2Ae%5E%281.05631725t%29
Now solve this for t just like we solved the earlier equation for k. You will get a decimal answer. Let's pretend you get t = 5.2117. Since 5.2117 is more than 5, the sales would reach 500000 more than 5 years after the start. IOW, the sales would reach 500000 sometime during the sixth year: 2006. (Note: 5.2117 is a pretend answer. You will not actually get 5.2117 for an answer.)