/*! 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 104 The formula \(A=25.1 e^{0.0187 t... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 104

The formula \(A=25.1 e^{0.0187 t}\) models the population of Texas, \(A\), in millions, \(t\) years after 2010 . a. What was the population of Texas in \(2010 ?\) b. When will the population of Texas reach 28 million?

Short Answer

Expert verified
a) The population of Texas in 2010 was 25.1 million. b) The population of Texas will reach 28 million approximately \(t\) years after 2010, where \(t\) is obtained from the formula \(t = \frac{ln(\frac{28}{25.1})}{0.0187}\).

Step by step solution

01

Find the population in 2010

Substitute \(t = 0\) into the formula: \(A = 25.1e^{0.0187 * 0} = 25.1\) million. Hence, the population of Texas in 2010 was 25.1 million.
02

Set up the equation to find when the population will reach 28 million

To find the time when the population reaches 28 million, set \(A = 28\) in the given formula: \(28 = 25.1e^{0.0187t}\).
03

Solve equation for \(t\)

First, divide both sides of the equation by 25.1 to isolate the exponent: \(\frac{28}{25.1} = e^{0.0187t}\). Then, take the natural logarithm of both sides: \(ln(\frac{28}{25.1}) = 0.0187t\). Finally, divide both sides by 0.0187 to solve for \(t\): \(t = \frac{ln(\frac{28}{25.1})}{0.0187}\). Calculate the value of \(t\) using a calculator.
04

Interpret the result

The value of \(t\) represents the number of years after 2010 when the population of Texas will reach 28 million. Round this value to the nearest year, because it does not make sense to have a fraction of a year.

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Most popular questions from this chapter

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation.. $$3^{x}=2 x+3$$

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Exercises \(153-155\) will help you prepare for the material covered in the next section. U.S. soldiers fight Russian troops who have invaded New York City. Incoming missiles from Russian submarines and warships ravage the Manhattan skyline. It's just another scenario for the multi-billion-dollar video games Call of Duty, which have sold more than 100 million games since the franchise's birth in 2003 The table shows the annual retail sales for Call of Duty video games from 2004 through 2010 . Create a scatter plot for the data. Based on the shape of the scatter plot, would a logarithmic function, an exponential function, or a linear function be the best choice for modeling the data? $$\begin{array}{cc} \hline \text { Year } & \begin{array}{c} \text { Retail Sales } \\ \text { (millions of dollars) } \end{array} \\ \hline 2004 & 56 \\ 2005 & 101 \\ 2006 & 196 \\ 2007 & 352 \\ 2008 & 436 \\ 2009 & 778 \\ 2010 & 980 \end{array}$$

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's power regression option to obtain a model of the form \(y=a x^{b}\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?
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