<< Chapter < Page | Chapter >> Page > |
A muskrat population oscillates 33 above and below average during the year, reaching the lowest value in January. The average population starts at 900 muskrats and increases by 7% each month. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$
A fish population oscillates 40 above and below average during the year, reaching the lowest value in January. The average population starts at 800 fish and increases by 4% each month. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$
$P(t)=-40\mathrm{cos}\left(\frac{\pi}{6}t\right)+800{(1.04)}^{t}$
A spring attached to the ceiling is pulled 10 cm down from equilibrium and released. The amplitude decreases by 15% each second. The spring oscillates 18 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.
A spring attached to the ceiling is pulled 7 cm down from equilibrium and released. The amplitude decreases by 11% each second. The spring oscillates 20 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.
$D(t)=7{\left(0.89\right)}^{t}\mathrm{cos}\left(40\pi t\right)$
A spring attached to the ceiling is pulled 17 cm down from equilibrium and released. After 3 seconds, the amplitude has decreased to 13 cm. The spring oscillates 14 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.
A spring attached to the ceiling is pulled 19 cm down from equilibrium and released. After 4 seconds, the amplitude has decreased to 14 cm. The spring oscillates 13 times each second. Find a function that models the distance, $\text{\hspace{0.17em}}D,$ the end of the spring is from equilibrium in terms of seconds, $\text{\hspace{0.17em}}t,$ since the spring was released.
$D(t)=19{\left(0.9265\right)}^{t}\mathrm{cos}\left(26\pi t\right)$
For the following exercises, create a function modeling the described behavior. Then, calculate the desired result using a calculator.
A certain lake currently has an average trout population of 20,000. The population naturally oscillates above and below average by 2,000 every year. This year, the lake was opened to fishermen. If fishermen catch 3,000 fish every year, how long will it take for the lake to have no more trout?
Whitefish populations are currently at 500 in a lake. The population naturally oscillates above and below by 25 each year. If humans overfish, taking 4% of the population every year, in how many years will the lake first have fewer than 200 whitefish?
$20.1\text{\hspace{0.17em}}$ years
A spring attached to a ceiling is pulled down 11 cm from equilibrium and released. After 2 seconds, the amplitude has decreased to 6 cm. The spring oscillates 8 times each second. Find when the spring first comes between $\text{\hspace{0.17em}}-0.1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0.1\text{cm,}$ effectively at rest.
A spring attached to a ceiling is pulled down 21 cm from equilibrium and released. After 6 seconds, the amplitude has decreased to 4 cm. The spring oscillates 20 times each second. Find when the spring first comes between $\text{\hspace{0.17em}}-0.1\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}0.1\text{cm,}$ effectively at rest.
17.8 seconds
Notification Switch
Would you like to follow the 'Precalculus' conversation and receive update notifications?