# Recent questions in Exponential models

Exponential models

### 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?

Exponential models

### Find the mean of the data. 9,12,11,11,10,7,4,8

Exponential models

### How can you determine whether a linear, exponential, or quadratic function best models data?

Exponential models

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

Exponential models

### Show that an exponential model fits the data. Then write a recursive rule that models the data.  $$\begin{array}{|c|c|} \hline n & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline f(n) & 2 & 6 & 18 & 54 & 162 & 486 \\ \hline \end{array}$$

Exponential models

### 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

Exponential models

### Give an example of an exponential function that models exponential growth and an example of an exponential function that models exponential decay.

Exponential models

### 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. $$\begin{array} \text{x} & \text{g(x)}\ \hline \text{−1−1} & \text{3}\ \text{0} & \text{6}\ \text{1} & \text{12}\ \text{2} & \text{18}\ \text{3} & \text{30}\ \end{array}$$

Exponential models

### Solve $$\displaystyle\frac{{{2}{x}-{4}}}{{3}}=\frac{{{4}{x}-{2}}}{{5}}$$

Exponential models

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

Exponential models

### Explain how to determine whether an exponential model of the form $$\displaystyle{y}={C}{e}^{{k}}{t}$$ models growth or decay.

Exponential models

### 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. $$\begin{matrix} \text{x} & \text{f(x)}\ \hline \text{−1−1} & \text{32 23}\ \text{0} & \text{3}\ \text{1} & \text{6}\ \text{2} & \text{12}\ \text{3} & \text{24}\ \end{matrix}$$\

Exponential models

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

Exponential models

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

Exponential models

### Use the two-way table of data from another student survey to answer the following question. $$\begin{array}{|c|c|} \hline & \text{Like Aerobic Excercise} \\ \hline & \text{Yes} & \text{No} & \text{Total} \\ \hline \text{Like Weight Lifting} \\ \hline \text{Yes} & 7 & 14 & 21 \\ \hline \text{No} & 12 & 7 & 19 \\ \hline \text{Total} & 19 & 21 & 40 \\ \hline \end{array}$$ Find the conditional relative frequency that a student likes to lift weights, given that the student likes aerobics.

Exponential models

### The exponential growth models describe the population of the indicated country, A, in millions, t years after 2006. Canada $$\displaystyle{A}={33.1}{e}^{{0.009}}{t}$$ Uganda $$\displaystyle{A}={28.2}{e}^{{0.034}}{t}$$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. Uganda's growth rate is approximately 3.8 times that of Canada's.

Exponential models

### 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.

Exponential models

### Find a recursive rule that models the exponential growth of $$\displaystyle{y}={2}{\left({1.08}\right)}^{{t}}$$

Exponential models