Problem 103 - Algebra and Trigonometry

Question

The formula \(A=37.3 e^{0.0095 t}\) models the population of California, \(A,\) in millions, \(t\) years after 2010 . a. What was the population of California in \(2010 ?\) b. When will the population of California reach 40 million?

Step-by-Step Solution

Answer

The population of California in 2010 was 37.3 million persons. The population of California will reach 40 million approximately 19 years after 2010, that is, around the year 2029.

Step 1: Find the population in 2010

To find the population in 2010, substitute \(t=0\) into the formula \(A = 37.3e^{0.0095t}\). This gives us \(A = 37.3e^{0.0095 * 0} = 37.3e^0 = 37.3\).

Step 2: Find the time when population reaches 40 million

To find the time period when A=40, substitute A=40 into the formula and solve for t. This gives \(40=37.3e^{0.0095t}\). To simplify this, first divide both sides by 37.3 to get \(\frac{40}{37.3}= e^{0.0095t}\). Then take the natural log of both sides to get \(ln\left(\frac{40}{37.3}\right)=0.0095t\). Finally, solve for t by dividing both sides by 0.0095.