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

The loudness L(x), measured in decibels (dB), of a sound of intensity x, measured in watts per square meter, is defined as $L(x)=10\log (\frac{x}{10})$, where $I_0={10}^{-12}$ watt per square meter is the least intense sound that a human ear can detected. Determine the loudness, in decibels, of each of the following sounds.
Heavy city traffic: intensity of $x={10}^{-3}$ watt per square meter.

Find an exponential function of the form ${\displaystyle P\left(t\right)=P_0{n}^{\frac{t}{T}}}$ that models the situation, and then find the equivalent exponential model of the form ${\displaystyle P\left(t\right)=P_0{e}^{rt}}$
1) Doubling time of 25 weeks, initial population of 1300

The annual sales S (in millions of dollars) for the Perrigo Company from 2004 through 2010 are shown in the table. $\begin{array}{|cccccccc|}\hline \text{Year}& 2004& 2005& 2006& 2007& 2008& 2009& 2010\\ \text{Sales, S}& 898.2& 1024.1& 1366.8& 1447.4& 1822.1& 2006.9& 2268.9\\ \hline\end{array}$

a) Use a graphing utility to create a scatter plot of the data. Let t represent the year, with t=4 corresponding to 2004. b) Use the regression feature of the graphing utility to find an exponential model for the data. Use the Inverse Property $b=e^{\ln b}$to rewrite the model as an exponential model in base e. c) Use the regression feature of the graphing utility to find a logarithmic model for the data. d) Use the exponential model in base e and the logarithmic model to predict sales in 2011. It is projected that sales in 2011 will be $2740 million. Do the predictions from the two models agree with this projection? Explain.

 

Andreas buys a new ailboat fo $15,540. He estimates that the boat will depreciate by 5% each year. Which exponential function models this situation?

Solve the given problem related to population growth.
During the first decade of this century the population of a certain city grew exponentially the population of the city was 146210 in 2000 and 217245 in 2010. Find the exponential growth function that models the population growth of the city use t= 0 to represent 2000 and t= 10 to represent 2010 and so on.
N(t)=?
Use your exponential growth function to predict the population of the city in 2016. Round to the nearest thousand.

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?

In addition to quadratic and exponential models, another common type of model is called a power model. Power models are models in the form ${\displaystyle \hat{y}=a\cdot x^p}$. 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. $\begin{array}{|ccc|}\hline \text{Planet}& \text{Distance from sun}\text{(astronomical units)}& \text{Period of revolution}\text{(Earth years)}\\ \text{Mercury}& 0.387& 0.241\\ \text{ Venus }& 0.723& 0.615\\ \text{ Earth }& 1.000& 1.000\\ \text{ Mars }& 1.524& 1.881\\ \text{ Jupiter }& 5.203& 11.862\\ \text{ Saturn }& 9.539& 29.456\\ \text{ Uranus }& 19.191& 84.070\\ \text{ Neptune }& 30.061& 164.810\\ \hline\end{array}$ Calculate and interpret the residual for Neptune.

The fox population in a certain region has a continuous growth rate of 6 percent per year. It is estimated that the population in the year 2000 was 18900.
(a) Find a function that models the population t years after 2000 (t=0 for 2000). Hint: Use an exponential function with base e.
(b) Use the function from part (a) to estimate the fox population in the year 2008.

Find the Polar, General Polar, Exponential and General Exponential Form of ${\displaystyle -5+9j}$

The half - life of a certain radioactive material is 85 days. An initial amount of the material has a mass of 801 kg Write an exponential function that models the decay of this material. Find how much radioactive material remains after 10 days. Round your answer to the nearest thousandth.

Sketch a graph of the function. Use transformations of functions whenever possible. ${\displaystyle f\left(x\right)=1-2x}$

The following table shows the approximate average household income in the United States in 1990, 1995, and 2003. ($t=0$ represents 1990.)
$\begin{array}{|cccc|}\hline \text{t(Year)}& 0& 5& 13\\ \text{H(Household Income in}\$1,000)& 30& 35& 43\\ \hline\end{array}$
Which of the following kinds of models would best fit the given data?
Explain your choice of model. ( a, b, c, and m are constants.)
a) Linear: $H(t)=mb+b$
b) Quadratic: $H(t)=a{t}^2+bt+c$
c) Exponential: $H(t)=A{b}^t$

Find Expontntial Model that fits the points shown in the graph of table

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 ${\displaystyle y=a{(1+r)}^t}$, that models the retail prrice of the automobile in 2014.
b. Use the model to predict the retail price of the automobile in 2014.

The original number of bacteria found present in a body was 160. Now, after a period of 9 hours, the body has a count of 920 bacteria. Write the exponential equation that models the growth of the bacteria.

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 ${\displaystyle \frac{dm}{dt}=-km}$ and the solution of that differential equation if ${\displaystyle m\left(t\right)=m_0{e}^{-kt}}$, 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 a calculator with a $y^x$ key or a key to solve: 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 $f(x)=574(1.026{)}^x$ models the population of India, f(x), in millions, x years after 1974.
a. Substitute 0 for x and, without using a calculator, find India’s population in 1974.
b. Substitute 27 for x and use your calculator to find India’s population, to the nearest million, in the year 2001 as modeled by this function.
c. Find India’s population, to the nearest million, in the year 2028 as predicted by this function.

Use exponential regression to find a function that models the data. $\begin{array}{|cccccc|}\hline x& 1& 2& 3& 4& 5\\ f(x)& 14& 7.1& 3.4& 1.8& 0.8\\ \hline\end{array}$

The exponential models describe the population of the indicated country, A, in millions, t years after 2010. Which countries have a decreasing population? By what percentage is the population of these countries decreasing each year?
Country B ${\displaystyle A=1193.1e^{0.006t}}$
Country C ${\displaystyle A=36.5e^{0.017t}}$
Country D ${\displaystyle A=121.7e^{-0.005t}}$
Country E ${\displaystyle A=145.3e^{-0.004t}}$

The following table lists the reported number of cases of infants born in the United States with HIV in recent years because their mother was infected.
Source:
Centers for Disease Control and Prevention.
$\begin{array}{|cc|}\hline \text{Year}& \text{Cases}\\ 1995& 295\\ 1997& 166\\ 1999& 109\\ 2001& 115\\ 2003& 94\\ 2005& 107\\ 2007& 79\\ \hline\end{array}$
a) Plot the data on a graphing calculator, letting ${\displaystyle t=0}$ correspond to the year 1995.
b) Using the regression feature on your calculator, find a quadratic, a cubic, and an exponential function that models this data.
c) Plot the three functions with the data on the same coordinate axes. Which function or functions best capture the behavior of the data over the years plotted?
d) Find the number of cases predicted by all three functions for 20152015. Which of these are realistic? Explain.