Problem 59 - College Algebra Essentials

Question

Suppose that a population that is growing exponentially increases from \(800,000\) people in 2007 to \(1,000,000\) people in \(2010 .\) Without showing the details, describe how to obtain the exponential growth function that models the data.

Step-by-Step Solution

Answer

The exponential growth function that models the data is given by \(P(t) = 800,000e^{t × \frac{ln(\frac{1,000,000}{800,000})}{3}}\) for \(t ≥ 0\).

Step 1: Understand the problem and identify given values

The problem is about exponential growth, which suggests using the formula \(P(t) = P_0 e^{rt}\). Here, \(P_0 = 800,000\) (the population in 2007), and \(P(t) = 1,000,000\) (the population in 2010). The time difference, \(t\), from 2007 to 2010 is 3 years.

Step 2: Substitute the values into the equation

Substitute 800,000 for \(P_0\), 1,000,000 for \(P(3)\), and 3 for \(t\) to get: 1,000,000 = 800,000e^{3r}.

Step 3: Solve for the growth rate \(r\)

First, divide both sides of the equation by 800,000 to isolate the exponential expression: \(\frac{1,000,000}{800,000} = e^{3r}\). From there, take the natural logarithm (ln) of both sides to solve for the exponent: \(ln(\frac{1,000,000}{800,000}) = 3r\). Therefore, the growth rate \(r = \frac{ln(\frac{1,000,000}{800,000})}{3}\)

Step 4: Form the exponential growth function

The last step is to substitute \(r\) back into the original growth formula, \(P(t) = P_0e^{rt}\), to obtain the function that represents the observed growth. This gives us the following final exponential growth function: \(P(t) = 800,000e^{t × \frac{ln(\frac{1,000,000}{800,000})}{3}}\) for \(t ≥ 0\).