# Use Exponential Models in Applications Elise invests $4500 in an account that compounds interest monthly and earns 6%. How long will it take for her money to double? Question Exponential models asked 2020-11-07 Use Exponential Models in Applications Elise invests$4500 in an account that compounds interest monthly and earns 6%. How long will it take for her money to double?

2020-11-08
Given:
$$\displaystyle{P}=\{4500}$$
$$\displaystyle{R}={6}\%$$
$$\displaystyle{A}=\{9000}$$
$$\displaystyle{n}=?$$
Since we are compounding monthly then we apply this formula
$$\displaystyle{A}={P}{\left({1}+{\frac{{{R}}}{{{12}{x}{100}}}}\right)}^{{{12}{n}}}$$
$$\displaystyle{9000}={4500}{\left({1}+{\frac{{{R}}}{{{12}{x}{100}}}}\right)}^{{{12}{n}}}$$
$$\displaystyle{2}={\left({1}+{.005}\right)}^{{{12}{n}}}{2}={1.005}^{{{12}{n}}}$$
Taking log on both side
$$\displaystyle{12}{n}{\log{{1.005}}}={\log{{2}}}$$
$$\displaystyle{12}{n}{x}{.0021}={0.301}$$
$$\displaystyle{n}={\frac{{{0.301}}}{{{0.026}}}}$$
$$\displaystyle{n}={11.57}$$ years

### Relevant Questions

Scientists are working with a sample of cobalt-56 in their laboratory. They begin with a sample that has 60 mg of cobalt-56, and they measure that after 31 days, the mass of cobalt-56 sample is 45.43 mg. Recall that the differential equation which models exponential decay is $$\frac{dm}{dt}=-km$$ and the solution of that differential equation if $$m(t)=m_0e^{-kt}$$, where $$m_0$$ is the initial mass and k is the relative decay rate.
a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
b) Use the information provided to determine the half-life of cobalt-56. Give your answer in days and round to the second decimal place. Show your calculation (do not just cite a formula).
c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
d) What will be the rate at which the mass is decaying when the sample has 50mg of cobalt-56? Make sure to indicate the appropriate units and round your answer to three decimal places.
Scientists are working with a sample of cobalt-56 in their laboratory. They begin with a sample that has 60 mg of cobalt-56, and they measure that after 31 days, the mass of cobalt-56 sample is 45.43 mg. Recall that the differential equation which models exponential decay is $$\displaystyle{\frac{{{d}{m}}}{{{\left.{d}{t}\right.}}}}=-{k}{m}$$ and the solution of that differential equation if $$\displaystyle{m}{\left({t}\right)}={m}_{{0}}{e}^{{-{k}{t}}}$$, where $$\displaystyle{m}_{{0}}$$ is the initial mass and k is the relative decay rate.
a) Use the information provided to compute the relative decay rate k. Show your calculation (do not just cit a formula).
b) Use the information provided to determine the half-life of cobalt-56. Give your answer in days and round to the second decimal place. Show your calculation (do not just cite a formula).
c) To the nearest day, how many days will it take for the initial sample of 60mg of cobalt-56 to decay to just 10mg of cobalt-56?
d) What will be the rate at which the mass is decaying when the sample has 50mg of cobalt-56? Make sure to indicate the appropriate units and round your answer to three decimal places.
Use Exponential Models in Applications
In the following exercises, solve.
Sayed deposits $20,000 in an investment account. What will be the value of his investment in 30 years if the investment is earning 7% per year and is compounded continuously? asked 2020-12-07 Suppose the retail of an automobile is$33,000 in 1999 and that it increases at 6% per year.
a. Write the equation of the exponential function, in the form $$\displaystyle{y}={a}{\left({1}+{r}\right)}^{{t}}$$, that models the retail prrice of the automobile in 2014.
b. Use the model to predict the retail price of the automobile in 2014.
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.
You deposit money into a saving account that earns interest. The balance Bo f the account after t years is given by $$\displaystyle{B}={1200}\cdot{1.05}^{{t}}{d}{o}{l}{l}{a}{r}{s}.$$
a. Make a table that shows the account balance for years 0 through 10.
b. You want to buy a home entertainment center that costs $1500. How long will you need to wait to have enough money in your savings account? Report your answer to the nearest whole year. asked 2021-02-13 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. asked 2021-01-24 A recently-installed machine earns the company revenueat a continuous rate of 60,000t + 45,000 dollars per year duringthe first six months of operation and at the continuous rate of75,000 dollars per year after the first six months. The cost of themachine is$150,000, the interest rate is 7% per year, compoundedcontinuously, and t is time in years since the machine wasinstalled.
(a)Find the present value of the revenue earned by the machineduring the first year of operation.
(b)Find how long it will take for the machine to pay for itself;that is, how long it will take for the present value of the revenueto equal the cost of the machine?
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.