4. The population of Californla was 20.0 million in 1970. By 1980 , the population had grown to 23.7 million. Assume that the population is growing exponentially. Show your work. 7 pts
Let
\[
\begin{aligned}
t & =\text { years since } 1970 \\
P(t) & =\text { population of Californla at time } t
\end{aligned}
\]
a) Find an exponental growth model for the population of California.
4. a)
b) Use the model to predict the population of California in the year 2010.
4. b)
c) Use the model to predict the year when the population of California will reach 50 million.

Step 1 ：We are given two data points: (0, 20) and (10, 23.7), where the first number in the pair is the number of years since 1970 and the second number is the population in millions. We are asked to find an exponential growth model, which has the form \(P(t) = P_0 * e^{kt}\), where \(P_0\) is the initial population, \(k\) is the growth rate, and \(t\) is time. We can use the two data points to solve for \(P_0\) and \(k\).

Step 2 ：From the given data, we can see that the initial population \(P_0\) is 20 million.

Step 3 ：We can use the second data point (10, 23.7) to solve for \(k\). Substituting \(P_0 = 20\), \(t = 10\), and \(P(t) = 23.7\) into the exponential growth model, we get \(23.7 = 20 * e^{10k}\). Solving this equation for \(k\), we get \(k = 0.01697\).

Step 4 ：Now we can write the exponential growth model as \(P(t) = 20 * e^{0.01697t}\). This model can be used to predict the population of California at any given year since 1970.

Step 5 ：\(\boxed{P(t) = 20 * e^{0.01697t}}\) is the exponential growth model for the population of California.