SOLUTION: The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. I

Question 268781: The population of a particular city is increasing at a rate proportional to its size. It follows the function P(t) = 1 + ke0.12t where k is a constant and t is the time in years. If the current population is 15,000, in how many years is the population expected to be 37,500? (Round to the nearest year.)
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

Before we try to answer the question we need to figure out what k is. We are given that the current population is 15,000. The "t" for "current" can be 0 so our equation for the current population would be:

This equation has only one unknown, k. So we can use this equation and some Algebra to solve for k. First we simplify:

Since :

Subtracting 1 from each side we get:

Now our function is

And we are finally ready to answer the question: "When will the population be 37,500?"

Subtract 1 from each side:

Divide both sides by 14999

Next we find the natural logarithm (ln) of each side:

which simplifies to

Next we divide both sides by 0.12:

This is an exact answer but it is hard to tell what number this is. So we'll get out our calculators and find a decimal approximation of the left side:

Since we're told to round to the nearest year, in approximately 8 years the population will be 37,500.

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