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Exponential models

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?

Exponential models

Use the exponential growth model $$A=A_0e^{kt}$$ 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?

Exponential models

Model the data using an exponential function $$\displaystyle{f{{\left({x}\right)}}}={A}{b}^{{x}}$$ $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{x}&{0}&{1}&{2}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{125}&{25}&{5}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$

Exponential models

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.) $$\displaystyle{\left({0},{16}\right)},{\left({60},{\frac{{{1}}}{{{4}}}}\right)}$$ $$\displaystyle{y}_{{1}}={C}{e}^{{{k}_{{1}}{t}}}$$ where C=? and $$\displaystyle{k}_{{1}}=?$$ $$\displaystyle{y}_{{2}}={C}{\left({2}\right)}^{{{k}_{{2}}{t}}}$$ where C=? and $$\displaystyle{k}_{{2}}=?$$

Exponential models

The population of California was 29.76 million in 1990 and 33.87 million in 2000. Assume that the population grows exponentially. (a) Find a function that models the population t years after 1990. (b) Find the time required for the population to double. (c) Use the function from part (a) to predict the population of California in the year 2010. Look up California’s actual population in 2010, and compare.

Exponential models

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)-?? $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{X}&-{2}&-{1}&{0}&{1}&{2}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{1.125}&{2.25}&{4.5}&{9}&{18}\backslash{h}{l}\in{e}{g{{\left({x}\right)}}}&{16}&{8}&{4}&{2}&{1}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$

Exponential models

Use the exponential growth model, $$A=A_0e^{kt}$$. In 1975, the population of Europe was 679 million. By 2015, the population had grown to 746 million. Solve, a. Find an exponential growth function that models the data for 1975 through 2015. b. By which year, to the nearest year, will the European population reach 800 million?

Exponential models

The number of users on a website has grown exponentially since its launch. After 2 months, there were 300 users. After 4 months there were 30000 users. Find the exponential function that models the number of users x months after the website was launched.

Exponential models

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?

Exponential models

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. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{c}\right|}{c}{\mid}\right\rbrace}{h}{l}\in{e}{x}&{1}&{2}&{3}&{4}&{5}&{6}&{7}&{8}&{9}&{10}\backslash{h}{l}\in{e}{f{{\left({x}\right)}}}&{409.4}&{260.7}&{170.4}&{110.6}&{74}&{44.7}&{32.4}&{19.5}&{12.7}&{8.1}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$

Exponential models

The initial value of a building is $986,260. After one year, the value of the building is$936,947. What exponential function models the expected value of the building? Estimate the value of the building after 3 years.

Exponential models

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: $$\displaystyle{V}={23},{300}{e}^{{-{0.311}{t}}}$$ Find the book values of the vehicle after year and after 3 years using each model.

Exponential models

Let V denote rainfall volume and W denote runoff volume (both in mm). According to the article “Runoff Quality Analysis of Urban Catchments with Analytical Probability Models” (J. of Water Resource Planning and Management, 2006: 4–14), the runoff volume will be 0 if $$\displaystyle{\left[{V}\ \leq\ {v}_{{d}}\ \right]}$$ and will $$\displaystyle{\left[{k}\ {\left({V}\ -\ {v}_{{d}}\right)}{\quad\text{if}\quad}\ {V}\ {>}\ {v}_{{d}}.\ {H}{e}{r}{e}\ {v}_{{d}}\right]}$$ is the volume of depression storage (a constant) and k (also a constant) is the runoff coefficient. The cited article proposes an exponential distribution with parameter $$\displaystyle{\left[\lambda\ {f}{\quad\text{or}\quad}\ {V}.\right]}$$ a. Obtain an expression for the cdf of W. [Note: W is neither purely continuous nor purely discrete, instead it has a “mixed” distribution with a discrete component at 0 and is continuous for values $$\displaystyle{w}{>}{0}$$.] b. What is the pdf of W for $$\displaystyle{w}{>}{0}$$? Use this to obtain an expression for the expected value of runoff volume.

Exponential models

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. $$\displaystyle{b}{e}{g}\in{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}{\left\lbrace{\left|{l}\right|}{l}{\left|{l}\right|}\right\rbrace}{h}{l}\in{e}{t}{\left(\text{in years}\right)}&{m}{\left(\text{amoun of radioactive material}\right)}\backslash{h}{l}\in{e}{0}&\backslash{h}{l}\in{e}{5730}\backslash{h}{l}\in{e}{11460}\backslash{h}{l}\in{e}{17190}\backslash{h}{l}\in{e}{e}{n}{d}{\left\lbrace{a}{r}{r}{a}{y}\right\rbrace}$$ 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.

Exponential models

A certain region had an initial rabbit population of 24, and the population is increasing at a rate of 26% each year due to a lack of natural predators. Write an exponential function that models N, the number of rabbits after t years, and use it to find the expected number of rabbits after 50 years.

Exponential models

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 $$\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{e}}}{N}\ {\left({8.86}\right)}.$$ 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 $$\displaystyle{\frac{{{d}{N}}}{{{\left.{d}{t}\right.}}}}={r}_{{{l}}}{N}\ {\left({1}\ -\ {\frac{{{N}}}{{{K}}}}\right)}\ {\left({8.88}\right)}$$ 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.

Exponential models

Newton's law of cooling indicates that the temperature of a warn object will decrease exponentially with time and will approach the temperature of the surrounding air. The temperature T(t) is modeled by $$\displaystyle{T}{\left({t}\right)}={T}_{{a}}+{\left({T}_{{0}}-{T}_{{a}}\right)}{e}^{{-{k}{t}}}$$. In this model, $$\displaystyle{T}_{{a}}$$ represents the temperature of the surrounding air, $$\displaystyle{T}_{{0}}$$ represents the initial temperature of the object and t is the time after the object starts cooling. The value of k is the cooling rate and is a constant related to the physical properties the object. A cake comes out of the oven at 335F and is placed on a cooling rack in a 70F kitchen. After checking the temperature several minutes later, it is determined that the cooling rate k is 0.050. Write a function that models the temperature T(t) of the cake t minutes after being removed from the oven.

Exponential models

The population of a small town was 3,600 people in the year 2015. The population increases by 4.5% every year. Write the exponential equation that models the population of the town t years after 2015. Then use your equation to estimate the population in the year 2025?

Exponential models