/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 103 The formula \(A=37.3 e^{0.0095 t... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 103

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?

Short Answer

Expert verified
a. The population of California in 2010 was 37.3 million. b. The population of California will reach 40 million approximately at the year found by evaluating \(\frac{\ln(\frac{40}{37.3})}{0.0095}\) years after 2010.

Step by step solution

01

Identify the Population in 2010

In the given exponential growth model, the value of the population at any given time \(t\) is given by: \(A=37.3 e^{0.0095 t}\). Here, \(t\) is the number of years after 2010. Therefore, the population in 2010 would be the population when \(t = 0\). Hence, calculate \(A\) when \(t = 0\). This leads to \(A = 37.3 e^{0.0095*0} = 37.3 e^{0}\). Since \(e^{0}\) equals 1, the population was 37.3 Million in 2010.
02

Calculate When the Population Reaches 40 Million

To find out when the population will reach 40 million, we need to solve for \(t\) in the model when \(A = 40\). This gives us: \(40 = 37.3 e^{0.0095 t}\). Start solving for \(t\) by firstly dividing both sides of the equation by 37.3 to isolate the exponential term: \(\frac{40}{37.3} = e^{0.0095 t}\). Now, take the natural logarithm (ln) of both sides to eliminate \(e\): \(\ln(\frac{40}{37.3}) = 0.0095 t\). Lastly, divide by 0.0095 to find the value of \(t\): \(t = \frac{\ln(\frac{40}{37.3})}{0.0095}\). Use a calculator to compute this.

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