SOLUTION: Hi. I've been having trouble with this problem, and the instructions aren't too clear to me. Here is the problem: In the year 1900, in the country Acirema, there were 100 Lawyer

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: Hi. I've been having trouble with this problem, and the instructions aren't too clear to me. Here is the problem: In the year 1900, in the country Acirema, there were 100 Lawyer      Log On


   

Question 712519: Hi. I've been having trouble with this problem, and the instructions aren't too clear to me. Here is the problem:
In the year 1900, in the country Acirema, there were 100 Lawyers and 5 million people. Every 10 years, the number of Lawyers doubles, and the population increases by 2 million. Let t be the number of years after 1900. Thus t=3 corresponds to 1903. Find the equation involving t whose solution tells you in which year 20 percent of the population are Lawyers. DO NOT SOLVE OR BEGIN TO SOLVE THIS EQUATION.
.2 = ______________

Thank You!
Answer by josgarithmetic(39626) About Me  (Show Source):
You can put this solution on YOUR website!
This solution is not perfect, but you can get a result with it, if you are willing to look at two different graphs, or to make two different graphs on the same coordinate axes.

The lawyers:
L = number of lawyers, doubles every 10 years.
L=L%5B0%5De%5E%28kt%29 use this to find k.
2L%5B0%5D=L%5B0%5De%5E%2810k%29, take natural log both sides,
ln%282%29=10k
k=%28ln%282%29%29%2F10
k=0.0693
Function is then+L=L%5B0%5De%5E%280.0693t%29
Or finally+L=100e%5E%280.0693t%29

The population:
P= poplulation starts at 5 million, increases by 2 million every 10 years.
t=10, pop is 5+2
t=20, pop is 5+2*2
t=30, pop is 5+3*2
Checking this pattern carefully, find that pop is 5%2B%281%2F10%29%2A2%2At
so P=5%2B%281%2F5%29t

The aim is to see for what time t L%2FP=0.20

We equivalently can write we want %28%28100e%5E%280.0693t%29%29%2F%285%2Bt%2F5%29%29=0.20
Which still may be written as highlight%28100e%5E%280.0693t%29=1%2Bt%2F25%29

I hope you don't mind some of the steps in the algebra in the above work are not shown.


So you want to graph the left hand member and the right hand member as functions on the same (or different) coordinate paper of system, and see where each crosses the other, or where intersect. No other symbolic algebraic way appears easy to see, so just make graphs and look for intersection.