Question 817148: There are currently 68 million cars in a certain country, increasingly by 1.8% annually. How many years will it take until this country has 81 million cars?
Answer by jsmallt9(3758)   (Show Source):
You can put this solution on YOUR website! A general equation which can be used for exponential growth (or decay) is:
where- "t" is the number of time units. (Units of time can be seconds, minutes, hours, days, years, etc.)
- "A" is the amount after t units of time have passed.
 is the initial amount (or the amount when t = 0).- "r" is a fraction, a decimal or a percent (written as a fraction or decimal) for the rate of change. Important: If this is a decay equation, "r" should be negative! The rate of change should use the same units of time as "t".
Since both the starting amount and the final amount of cars is expressed in the millions, I'm going to say that the A's are in millions of cars so I can just write 68 and 81 instead of 68000000 and 81000000. With this the equation for this problem would be:
 (where 0.018 is 1.8% in decimal form)
This simplifies to:
Now we can use this to find the answer to our problem. We want to find "t" for A = 81:
To solve this, first isolate the base and its exponent. Dividing both sides by 68:
Next we find the logarithm of each side. Any base of logarithm may be used. But choosing certain bases can make things a little easier:- Matching the base of the logarithm to the base of the exponent will result in the simplest possible expression for the exact solution.
- Choosing a base of logarithm that your calculator "knows" (base 10 ("log") and/or base e ("ln") are on many calculators) will result in an expression that can easily be converted to a decimal approximation. (With word problems, this is the usual choice.)
Since this is a word problem I will choose base e logarithms:
Next we use a property of logarithms,  , to "move" the exponent out in front. (It is this very property that is the reason for using logarithms in the first place. It lets us move the exponent, where the variable is, to a location, in front, where we can "get at it" with "regular" algebra.) Using this property on our equation we get:
Next divide by ln(1.018):
This is an exact expression for the solution to the problem. But we probably want a number so we will use our calculators to figure out an approximate answer. First we'll find the logarithms. (If your calculator does not have parentheses buttons, then you probably need to change 81/68 into a decimal first.) Be sure to find the two logs and then divide. You cannot divide 81/68 by 1.018 first!!)
 (rounded to four places)
 (rounded to four places)
So it will take nearly 10 years for there to be 81 million cars.
|
|
|
