Skip to main content
Contents Dark Mode Prev Up Next \(
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\definecolor{fillinmathshade}{gray}{0.9}
\newcommand{\fillinmath}[1]{\mathchoice{\colorbox{fillinmathshade}{$\displaystyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\textstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptstyle \phantom{\,#1\,}$}}{\colorbox{fillinmathshade}{$\scriptscriptstyle\phantom{\,#1\,}$}}}
\)
Subsection 2.2 Exponential Modeling Exercises
The population of a small town has been growing. The population is shown in
Table 2.2.8 .
Table 2.2.8. Population, \(P(t)\)
\(P(t)\) 1000
1344
1806
2427
What class of functions (linear, quadratic or exponential) would be best to model the population of the town. Be sure to explain your reasoning.
What is the average number of people added to the town per year between 1980 and 1990?
Assuming that the population continues to grow exponentially, write a function equation for
\(P(t)\text{.}\)
If the population continues to grow according to the model, what is the population in 2015?
When will the population reach 5,000?
The pesticide DDT was used in the US and later banned. The half-life of DDT is about 15 years.
Write an exponential model for the amount of DDT,
\(A(t)\text{,}\) remaining after
\(t\) years, if the initial sample is 100 grams.
According to your model, how much of the initial sample will remain after 60 years?
How many years will it take for the sample to decay to 1 gram?
Plutonium 238 is a radioactive element that decays at a rate of
\(0.8\%\) per year.
What percentage of an initial supply of 500 grams Plutonium 238 will remain after 40 years?
How many years will it be until an initial supply of 500 grams of Plutonium 238 has decayed to half of its initial mass?
When buying a new car, one consideration is how fast the car loses value, known as the depreciation rate. For example, one version of the Jeep Liberty loses value more rapidly than some of its competitors. From the initial purchase price of $23,395, an owner can expect the value of the car 5 years later to be $15,239.
What is the average dollar value decline per year during the first 5 years of ownership?
Use an exponential model produce a function that gives the value of the Jeep in terms of the number of years,
\(t\text{,}\) since the Jeep was new.
Use your model to predict the value of the Jeep when it was 3 years old.
When should the value of the Jeep decline to $5,000?
A river where salmon spawn had 2214 salmon spawn in 2015. In 2018, only 2023 salmon spawned in the same river.
Assuming the number of salmon spawning is decreasing exponentially, write a formula for the number of salmon spawning in the river
\(t\) years after 2015.
If the number of salmon spawning in the river continues to decrease exponentially, how many salmon will spawn in the river in 2025?
If the number of salmon spawning in the river continues to decrease exponentially, in what year will the number of salmon spawning in the river decrease to half of the number of 2015?
Oscar charges $5 per linear foot to paint any standard outdoor wooden fence, plus $20 to cover incidental items, such as brushes.
Write a function equation that gives the cost to paint a fence of length
\(L\) feet.
What is the cost to paint a fence 50 feet long?
If Oscar recently painted a fence and charged $110, how long was the fence?
For a certain species of shark, its length (in feet) varies according to the equation
\begin{equation*}
l(w)=kw^\frac{1}{3}
\end{equation*}
where \(w\) is the shark’s weight in pounds.
If a 6 foot shark weighs 200 pounds, write an equation for
\(l(w)\text{.}\)
Using your equation from
Item 7.a , how long will a shark be if it weighs 400 pounds?
Using your equation from
Item 7.a , how much will a shark weigh if it is 4 feet long?
Let
\(f(x)=-x^2+4x\) and
\(g(x)=x-4\text{.}\)
