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

The values of two functions, f and g, are given in a table. One, both, or neither of them may be exponential. Give the exponential models for those that are.
f(x)-?
g(x)-??
$\begin{array}{|llllll|}\hline X& -2& -1& 0& 1& 2\\ f(x)& 1.125& 2.25& 4.5& 9& 18\\ g(x)& 16& 8& 4& 2& 1\\ \hline\end{array}$

The population of a region is growing exponentially. There were 10 million people in 1980 (when $t=0$) and 75 million people in 1990. Find an exponential model for the population (in millions of people) at any time tt, in years after 1980.
P(t)=?
What population do you predict for the year 2000?
Predicted population in the year 2000 =million people. What is the doubling time?

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.
$\begin{array}{|ccccccccccc|}\hline x& 1& 2& 3& 4& 5& 6& 7& 8& 9& 10\\ f(x)& 409.4& 260.7& 170.4& 110.6& 74& 44.7& 32.4& 19.5& 12.7& 8.1\\ \hline\end{array}$

A sport utility vehicle that costs $23,300 new has a book value of $12500 after 2 years.
Linear model: $V=-5400t+23300$
Exponential model: ${\displaystyle V=23,300e^{-0.311t}}$
Find the book values of the vehicle after year and after 3 years using each model.

The burial cloth of an Egyptian mummy is estimated to contain 560 g of the radioactive materialcarbon-14, which has a half life of 5730 years.
a. Complete the table below. Make sure you justify your answer by showing all the steps.
$\begin{array}{|ll|}\hline t(\text{in years})& m(\text{amoun of radioactive material})\\ 0& \\ 5730\\ 11460\\ 17190\\ \hline\end{array}$
b. Find an exponential function that models the amount of carbon-14 in the cloth, y, after t years. Make sure you justify your answer by showing all the steps.
c. If the burial cloth is estimated to contain 49.5% of the original amount of carbon-14, how long ago was the mummy buried. Give exact answer. Make sure you justify your answer by showing all the steps.

Several models have been proposed to explain the diversification of life during geological periods. According to Benton (1997), The diversification of marine families in the past 600 million years (Myr) appears to have followed two or three logistic curves, with equilibrium levels that lasted for up to 200 Myr. In contrast, continental organisms clearly show an exponential pattern of diversification, and although it is not clear whether the empirical diversification patterns are real or are artifacts of a poor fossil record, the latter explanation seems unlikely. In this problem, we will investigate three models fordiversification. They are analogous to models for populationgrowth, however, the quantities involved have a differentinterpretation. We denote by N(t) the diversification function,which counts the number of taxa as a function of time, and by rthe intrinsic rate of diversification.
(a) (Exponential Model) This model is described by ${\displaystyle \frac{dN}{dt}=r_{e}N\left(8.86\right).}$ Solve (8.86) with the initial condition N(0) at time 0, and show that ${\displaystyle r_e}$ can be estimated from ${\displaystyle r_e=\frac{1}{t}\ln \left[\frac{N\left(t\right)}{N\left(0\right)}\right]\left(8.87\right)}$
(b) (Logistic Growth) This model is described by ${\displaystyle \frac{dN}{dt}=r_{l}N(1-\frac{N}{K})\left(8.88\right)}$ where K is the equilibrium value. Solve (8.88) with the initial condition N(0) at time 0, and show that ${\displaystyle r_l}$ can be estimated from ${\displaystyle r_l=\frac{1}{t}\ln \left[\frac{K-N\left(0\right)}{N\left(0\right)}\right]+\frac{1}{t}\ln \left[\frac{N\left(t\right)}{K-N\left(t\right)}\right]\left(8.89\right)}$ for ${\displaystyle N\left(t\right)<K.}$
(c) Assume that ${\displaystyle N\left(0\right)=1}$ and ${\displaystyle N\left(10\right)=1000.}$ Estimate ${\displaystyle r_e}$ and ${\displaystyle r_l}$ for both ${\displaystyle K=1001}$ and ${\displaystyle K=10000.}$
(d) Use your answer in (c) to explain the following quote from Stanley (1979): There must be a general tendency for calculated values of ${\displaystyle \left[r\right]}$ to represent underestimates of exponential rates,because some radiation will have followed distinctly sigmoid paths during the interval evaluated.
(e) Explain why the exponential model is a good approximation to the logistic model when ${\displaystyle \frac{N}{K}}$ is small compared with 1.

The table gives the number of active Twitter users worldwide, semiannually from 2010 to 2016. $$\begin{array}{|cccc|}\hline \text{Years since}& \text{January 1, 2010}& \text{Twitter user}& \text{(millions)}\\ 0& 30& 3.5& 232\\ 0.5& 49& 4.0& 255\\ 1.0& 68& 4.5& 284\\ 1.5& 101& 5.0& 302\\ 2.0& 138& 5.5& 307\\ 2.5& 167& 6.0& 310\\ 3.0& 204& 6.5& 317\\ \hline\end{array}$$

Use a calculator or computer to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models.

In 2015, there were 3,600 people living in a small town. Every year, the population grows by 4.5%. Write the exponential equation that represents the town's population t years from now, in 2015. Then, how about using your equation to predict the population in 2025?

The table gives the midyear population of Japan, in thousands, from 1960 to 2010.
$\begin{array}{|cc|}\hline \text{Year}& \text{Population}\\ 1960& 94.092\\ 1965& 98.883\\ 1970& 104.345\\ 1975& 111.573\\ 1980& 116.807\\ 1985& 120.754\\ 1990& 123.537\\ 1995& 125.327\\ 2000& 126.776\\ 2005& 127.715\\ 2010& 127.579\\ \hline\end{array}$
Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose $t=0$ to correspond to 1960 or 1980.]

Consider the exponential function $f(x)=1,250\cdot {1.08}^x$ which models the amount of money invested in a bond fund where x represents the number of years since the money was invested. a. What is the rate of growth or decay?
b. What does 1,250 represent?