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

Suppose that a population that is growing exponentially increases from 800,000 people in 2007 to 1,000,000 people in 2010. Without showing the details, describe how to obtain the exponential growth function that models the data.

The population of Williston, North Dakota, has grown rapidly over the past decade due to an oil boom. The table gives the population of the town in 2007, 2009, and 2011. Years Since 20007811 Population (thousands)12.413.016.0
Use your calculator to find an exponential equation that models the growth of the town.

The table shows the number of employees at a software company in various years.
Years Since 2000 4 6 8 10 Number of Employees 32 40 75 124
Use your calculator to find an exponential equation that models the growth of the company.

Find the exponential function whose grapgh is given

The popularity of fads and fashions often decays exponentially. One example is ticket sales for a popular movie. The table shows the total money spent per weekend on tickets in the United States and Canada for the movie The Da Vinci Code. $$$$\begin{array}{cc}\text{Weekend in 2006}& TicketSales(millions)\\ \text{May 19—May 21}& \text{77.1}\\ \text{May 26—May 28}& \text{34.0}\\ \text{June 2—June 4}& \text{18.6}\\ \text{June 9—June 11}& \text{10.4}\\ \text{June 16—June 18}& \text{5.3}\\ \text{June 23—June 25}& \text{4.1}\\ \text{June 30—July 2}& \text{2.3}\end{array}$$$ a) Use a graphing calculator to create a scatter plot of the data. b) Draw a quadratic curve of best fit. - Press STAT, cursor over to display the CALC menu, and select 5:QuadReg. - Press VARS, and cursor over to display the Y-VARS menu. Select 1:Function and then select 1:Y1. - Press ENTER to get the QuadReg screen, and press GRAPH. c) Draw an exponential curve of best fit. - Press STAT, cursor over to display the CALC menu, and select 0:ExpReg. - Press VARS, and cursor over to display the Y-VARS menu. Select 1:Function and then select 2:Y2. - Press ENTER to get the ExpReg screen, and press GRAPH. d) Examine the two curves. Which curve of best fit best models the data?

When a < 0 and b>1,y=abx models negative exponential growth. a. Write an exponential function that models negative growth. b. Give an example of a situation that could be modeled by your function. c. Explain one difference between negative exponential growth and exponential decay.

What function models exponential decay?
A. ${\displaystyle y=-3{\left(\frac{1}{2}\right)}^x}$
B. ${\displaystyle y=-\left(\frac{1}{2}\right){\left(3\right)}^x}$
C. ${\displaystyle y=\left(\frac{1}{2}\right){\left(3\right)}^x}$
D. ${\displaystyle y=3{\left(\frac{1}{2}\right)}^x}$

Show that an exponential model fits the data. Then write a recursive rule that models the data.

A computer valued at $1500 loses 20% of its value each year. a. Write a function rule that models the value of the computer. b. Find the value of the computer after 3 yr. c. In how many years will the value of the computer be less than $500? Remember the general form of an exponential function: $f(x)=a\ast b^x$. Given your table and situation, determine values for a and b. Write the appropriate exponential function.

Explain when the function ${\displaystyle y=a\cdot b^x}$ models exponential growth and when it models exponential decay.

Find a recursive rule that models the exponential decay of ${\displaystyle y=1600{\left(0.97\right)}^t}$

Biff observes that in every math test he has taken this year, he has scored 2 points higher than the previous test. His score on the first test was 56. He models his test scores with the exponential function ${\displaystyle s\left(n\right)=28\cdot 2^n}$ where s(n) is the score on his nth test. Is this a reasonable model? Explain.

A share of ABC stock was worth $60 in 2005 and only worth $45 in 2010. a. Find the multiplier and the percent decrease. b. Write an exponential function that models the value of the stock starting from 2005. c. Assuming that the decline in value continues at the same rate , use your answer to (b) to predict the value in 2020.

You decide to start saving pennies according to the following pattern. You save 1 penny the first day, 2 pennies the second day, 4 the third day, 8 the fourth day, and so on. Each day you save twice the number of pennies you saved on the previous day. Write an exponential function that models this problem. How many pennies do you save on the thirtieth day?

In addition to quadratic and exponential models, another common type of model is called a power model. Power models are models in the form $\hat{y}=a\times x^p$. Here are data on the eight planets of our solar system. Distance from the sun is measured in astronomical units (AU), the average distance Earth is from the sun.
$\begin{array}{|lcc|}\hline \text{ Planet }& \begin{array}{c}\text{ Distance from sun }\\ \text{ (astronomical units) }\end{array}& \begin{array}{c}\text{ Period of revolution }\\ \text{ (Earth years) }\end{array}\\ \text{ Mercury }& 0.387& 0.241\\ \text{ Venus }& 0.723& 0.615\\ \text{ Earth }& 1.000& 1.000\\ \text{ Mars }& 1.524& 1.881\\ \text{ Jupiter }& 5.203& 11.862\\ \text{ Saturn }& 9.539& 29.456\\ \text{ Uranus }& 19.191& 84.070\\ \text{ Neptune }& 30.061& 164.810\\ \hline\end{array}$
Use the PwrReg command on your graphing calculator to find a power model relating $y=$ period to $x=$ distance.

Determine whether the function given by the table is linear, exponential, or neither. If the function is linear, find a linear function that models the data. If it is exponential, find an exponential function that models the data. x f(x) -1 8/7 0 8 1 56 2 392

The exponential function $f(x)=42.2(1.56{)}^x$ models the average amount spent, f(x), in dollars, at a shopping mall after x hours. What is the average amount spent, to the nearest dollar, after four hours?

Use the exponential growth model $A=A_0{e}^{kt}$ to solve: In 2000, there were 110 million cellphone subscribers in the United States. By 2010, there were 303 million subscribers
a. Find the exponential function that models the data.
b. According to the model, in which year were there 400 million cellphone subscribers in the United States?

Model the data using an exponential function $f(x)=A{b}^x$
$\begin{array}{|llll|}\hline x& 0& 1& 2\\ f(x)& 125& 25& 5\\ \hline\end{array}$

Find exponential models
${\displaystyle y_1=C{e}^{k_{1}t}}$
and
${\displaystyle y_2=C{\left(2\right)}^{k_{2}t}}$
That pass through the two given points. Compare the values of ${\displaystyle k_1}$ and ${\displaystyle k_2}$. (If you round your answer, round to four decimal places.)
${\displaystyle (0,16),(60,\frac{1}{4})}$
${\displaystyle y_1=C{e}^{k_{1}t}}$ where C=?
and ${\displaystyle k_1=?}$
${\displaystyle y_2=C{\left(2\right)}^{k_{2}t}}$ where C=?
and ${\displaystyle k_2=?}$