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For the following exercises, identify whether the statement represents an exponential function. Explain.
The average annual population increase of a pack of wolves is 25.
A population of bacteria decreases by a factor of $\text{\hspace{0.17em}}\frac{1}{8}\text{\hspace{0.17em}}$ every $\text{\hspace{0.17em}}24\text{\hspace{0.17em}}$ hours.
exponential; the population decreases by a proportional rate. .
The value of a coin collection has increased by $\text{\hspace{0.17em}}3.25\%\text{\hspace{0.17em}}$ annually over the last $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years.
For each training session, a personal trainer charges his clients $\text{\hspace{0.17em}}\text{\$}5\text{\hspace{0.17em}}$ less than the previous training session.
not exponential; the charge decreases by a constant amount each visit, so the statement represents a linear function. .
The height of a projectile at time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is represented by the function $\text{\hspace{0.17em}}h(t)=-4.9{t}^{2}+18t+40.$
For the following exercises, consider this scenario: For each year $\text{\hspace{0.17em}}t,$ the population of a forest of trees is represented by the function $\text{\hspace{0.17em}}A(t)=115{(1.025)}^{t}.\text{\hspace{0.17em}}$ In a neighboring forest, the population of the same type of tree is represented by the function $\text{\hspace{0.17em}}B(t)=82{(1.029)}^{t}.\text{\hspace{0.17em}}$ (Round answers to the nearest whole number.)
Which forest’s population is growing at a faster rate?
The forest represented by the function $\text{\hspace{0.17em}}B(t)=82{(1.029)}^{t}.$
Which forest had a greater number of trees initially? By how many?
Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $\text{\hspace{0.17em}}20\text{\hspace{0.17em}}$ years? By how many?
After $\text{\hspace{0.17em}}t=20\text{\hspace{0.17em}}$ years, forest A will have $\text{\hspace{0.17em}}43\text{\hspace{0.17em}}$ more trees than forest B.
Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after $\text{\hspace{0.17em}}100\text{\hspace{0.17em}}$ years? By how many?
Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?
Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but because forest B actually grows at a faster rate, the population will eventually become larger than forest A and will remain that way as long as the population growth models hold. Some factors that might influence the long-term validity of the exponential growth model are drought, an epidemic that culls the population, and other environmental and biological factors.
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
$y=300{\left(1-t\right)}^{5}$
$y=220{\left(1.06\right)}^{x}$
exponential growth; The growth factor, $\text{\hspace{0.17em}}1.06,$ is greater than $\text{\hspace{0.17em}}1.$
$y=16.5{\left(1.025\right)}^{\frac{1}{x}}$
$y=11,701{\left(0.97\right)}^{t}$
exponential decay; The decay factor, $\text{\hspace{0.17em}}0.97,$ is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}1.$
For the following exercises, find the formula for an exponential function that passes through the two points given.
$\left(0,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(3,750)$
$\left(0,2000\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(2,20)$
$f(x)=2000{(0.1)}^{x}$
$\left(-1,\frac{3}{2}\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,24\right)$
$\left(-2,6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\left(3,1\right)$
$f(x)={\left(\frac{1}{6}\right)}^{-\frac{3}{5}}{\left(\frac{1}{6}\right)}^{\frac{x}{5}}\approx 2.93{\left(0.699\right)}^{x}$
$\left(3,1\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}(5,4)$
For the following exercises, determine whether the table could represent a function that is linear, exponential, or neither. If it appears to be exponential, find a function that passes through the points.
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