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Chapter 4: Problem 37

Housing Costs Average annual per-household spending on housing over the years \(1990-2008\) is approximated by $$ H=8790 e^{0.0382 t} $$ where \(t\) is the number of years since \(1990 .\) Find \(H\) to the nearest dollar for each year. (Source: U.S. Bureau of Labor Statistics.) (a) 2000 (b) 2005 (c) 2008

Short Answer

Expert verified
In 2000: \( H \approx 12{,}875 \). In 2005: \( H \approx 15{,}600 \). In 2008: \( H \approx 17{,}488 \).

Step by step solution

01

Identify the given formula

The given formula for annual per-household spending on housing is: \[ H = 8790 e^{0.0382 t} \] where \( t \) is the number of years since 1990.
02

Calculate \( t \) for each year

Calculate the value of \( t \) for each of the given years.(a) For the year 2000: \[ t = 2000 - 1990 = 10 \](b) For the year 2005: \[ t = 2005 - 1990 = 15 \](c) For the year 2008: \[ t = 2008 - 1990 = 18 \]
03

Find \( H \) for the year 2000

Substitute \( t = 10 \) into the formula and calculate \( H \):\[ H = 8790 e^{0.0382 \times 10} \]Calculate:\[ H \approx 8790 e^{0.382} \approx 8790 \times 1.465 \approx 12{,}875 \]So, \( H \approx 12{,}875 \).
04

Find \( H \) for the year 2005

Substitute \( t = 15 \) into the formula and calculate \( H \):\[ H = 8790 e^{0.0382 \times 15} \]Calculate:\[ H \approx 8790 e^{0.573} \approx 8790 \times 1.774 \approx 15{,}600 \]So, \( H \approx 15{,}600 \).
05

Find \( H \) for the year 2008

Substitute \( t = 18 \) into the formula and calculate \( H \):\[ H = 8790 e^{0.0382 \times 18} \]Calculate:\[ H \approx 8790 e^{0.688} \approx 8790 \times 1.989 \approx 17{,}488 \]So, \( H \approx 17{,}488 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

precalculus applications
Precalculus is a branch of mathematics that helps in understanding concepts needed for calculus. It involves the study of functions, limits, and other mathematical models to describe real-world phenomena. One practical application is in economics, where mathematical models are used to predict future trends. In our exercise, we are given a model to estimate annual housing costs over several years. By plugging values into this model, students can see how precalculus tools are used to solve real-life problems.
exponential functions
Exponential functions are fundamental in math and science. They are expressed in the form: \[ f(x) = a \times e^{bx} \] where 'a' and 'b' are constants, and 'e' is the base of natural logarithms (approximately 2.71828). The rate of change of the function is proportional to its current value, which makes exponential functions ideal for modeling growth processes. For example, in our given equation \[ H = 8790 \times e^{0.0382t} \] we see an exponential function showing how housing costs increase over time. By understanding and using exponential functions, students can predict growth patterns in various fields, including finance, population studies, and natural sciences.
housing cost trends
Understanding housing cost trends is crucial for making informed financial decisions. Housing prices can be influenced by economic factors like inflation and market demand. The given exercise shows how annual housing costs change exponentially. By calculating the respective costs for specific years:
  • 2000: \( H \approx 12,875 \)
  • 2005: \( H \approx 15,600 \)
  • 2008: \( H \approx 17,488 \)
students can see the power of these trends. Such knowledge is invaluable for real estate investment, budgeting, and economic planning. Recognizing these patterns helps individuals and businesses to anticipate future expenses and plan accordingly.

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Most popular questions from this chapter

If the function is one-to-one, find its inverse. $$\\{(1,-3),(2,-7),(4,-3),(5,-5)\\}$$

Use the various properties of exponential and logarithmic functions to evaluate the expressions in parts \((a)-(c)\) Given \(g(x)=e^{x},\) find (a) \(g(\ln 4) \quad\) (b) \(g\left(\ln \left(5^{2}\right)\right)\) (c) \(g\left(\ln \left(\frac{1}{\varepsilon}\right)\right)\)

The table shows the cost of a year's tuition, room and board, and fees at 4 -year public colleges for the years \(2002-2010 .\) Letting \(y\) represent the cost and \(x\) the number of years since 2002 we find that the function $$f(x)=9318(1.06)^{x}$$ models the data quite well. According to this function, when will the cost in 2002 be doubled? $$\begin{array}{c|c}\text { Year } & \text { Average Annual cost } \\\\\hline 2002 & \$ 9,119 \\ \hline 2003 & \$ 9,951 \\\\\hline 2004 & \$ 10,648 \\\\\hline 2005 & \$ 11,322 \\\ \hline 2006 & \$ 11,929 \\\\\hline 2007 & \$ 12,698 \\\\\hline 2008 & \$ 13,467 \\ \hline 2009 & \$ 14,265 \\\\\hline 2010 & \$ 15,067 \\\\\hline\end{array}$$

For each function as defined that is one-to-one, (a) write an equation for the inverse function in the form \(y=f^{-1}(x),\) (b) graph \(f\) and \(f^{-1}\) on the same axes, and \((c)\) give the domain and the range of \(f\) and \(f^{-1}\). If the function is not one-to-one, say so. $$f(x)=\frac{-3 x+12}{x-6}, \quad x \neq 6$$

Let \(u=\ln a\) and \(v=\ln b .\) Write each expression in terms of \(u\) and \(v\) without using the In function. $$\ln \left(\sqrt[3]{a} \cdot b^{4}\right)$$
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