Get In-Depth Help with Exponential models Solutions and Real-World Applications

For the following exercises, use a graphing utility to create a scatter diagram of the data given in the table. Observe the shape of the scatter diagram to determine whether the data is best described by an exponential, logarithmic, or logistic model. Then use the appropriate regression feature to find an equation that models the data. When necessary, round values to five decimal places.
$\begin{array}{|ccccccccccc|}\hline x& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ f(x)& 20& 21.6& 29.2& 36.4& 46.6& 55.7& 72.6& 87.1& 107.2& 138.1\\ \hline\end{array}$

Sociologists have found that information spreads among a population at an exponential rate. Suppose that the function $y=525(1-e^{-0.038t})$ models the number of people in a town of 525 people who have heard news within t hours of its distribution. How many people will have heard about the opening of a new grocery store within 24 hours of the announcement?

How do you tell which function—linear, exponential, or quadratic—best represents data?

Show that an exponential model fits the data. Then write a recursive rule that models the data.
$\begin{array}{|ccccccc|}\hline n& 0& 1& 2& 3& 4& 5\\ f(n)& 162& 54& 18& 6& 2& \frac{2}{3}\\ \hline\end{array}$

Mr. Lopez picks up 2 pallets of dog food to deliver to the animal shelter. Each pallet contains 40 bags of dog food, and the dog food weighs $37\frac{1}{2}$ pounds per bag. What is the total weight, in tons, of the dog food Mr. Lopez delivers to the animal shelter?
A $1\frac{1}{2}$ T
B 2 T
C $2\frac{1}{2}$ T
D $\frac{3}{4}$ T

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data.

$\begin{array}{}x& \text{g(x)}\\ \text{-1,-1}& \text{3}\\ \text{0}& \text{6}\\ \text{1}& \text{12}\\ \text{2}& \text{18}\\ \text{3}& \text{30}\end{array}$

The populations P (in thousands) of Luxembourg for the years 1999 through 2013 are shown in the table, where t represents the year, with $t=9$ corresponding to 1999.

$\begin{array}{|lc|}\hline \text{ Year }& \text{ Population, }P\\ 1999& 427.4\\ 2000& 433.6\\ 2001& 439.0\\ 2002& 444.1\\ 2003& 448.3\\ 2004& 455.0\\ 2005& 461.2\\ 2006& 469.1\\ 2007& 476.2\\ 2008& 483.8\\ 2009& 493.5\\ 2010& 502.1\\ 2011& 511.8\\ 2012& 524.9\\ 2013& 537.0\\ \hline\end{array}$

(a) Use the regression feature of a graphing utility to find a linear model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (b) Use the regression feature of the graphing utility to find a power model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (c) Use the regression feature of the graphing utility to find an exponential model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (d) Use the regression feature of the graphing utility to find a logarithmic model for the data and to identify the coefficient of determination. Plot the model and the data in the same viewing window. (e) Which model is the best fit for the data? Explain. (f) Use each model to predict the populations of Luxembourg for the years 2014 through 2018. (g) Which model is the best choice for predicting the future population of Luxembourg? Explain. (h) Were your choices of models the same for parts (e) and (g)? If not, explain why your choices were different.

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data.

$\begin{array}{|cccccc|}\hline x& -1& 0& 1& 2& 3\\ g(x)& 2& 5& 8& 11& 14\\ \hline\end{array}$

Determine whether the given function is linear, exponential, or neither. For those that are linear functions, find a linear function that models the data; for those that are exponential, find an exponential function that models the data.

$\begin{array}{cc}\text{x}& \text{f(x)}\\ \text{-1,-1}& \text{32, 23}\\ \text{0}& \text{3}\\ \text{1}& \text{6}\\ \text{2}& \text{12}\\ \text{3}& \text{24}\end{array}$

Use the factors to identify the zeros of $f(x)=3x^3+12x^2-36x$. Then sketch the graph of the polynomial.

India is currently one of the world’s fastest-growing countries. By 2040, the population of India will be larger than the population of China; by 2050, nearly one-third of the world’s population will live in these two countries alone. The exponential function ${\displaystyle f\left(x\right)=574{\left(1.026\right)}^x}$ models the population of India, f(x), in millions, x years after 1974. Find India’s population, to the nearest million, in the year 2055 as predicted by this function.