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The sea ice area around the South Pole fluctuates between about 18 million square kilometers in September to 3 million square kilometers in March. Assuming a sinusoidal fluctuation, when are there more than 15 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.
From July 8 to October 23
During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. If the rainfall fluctuates between a low of 2 inches on day 10 and 12 inches on day 55, during what period is daily rainfall more than 10 inches?
During a 90-day monsoon season, daily rainfall can be modeled by sinusoidal functions. A low of 4 inches of rainfall was recorded on day 30, and overall the average daily rainfall was 8 inches. During what period was daily rainfall less than 5 inches?
From day 19 through day 40
In a certain region, monthly precipitation peaks at 8 inches on June 1 and falls to a low of 1 inch on December 1. Identify the periods when the region is under flood conditions (greater than 7 inches) and drought conditions (less than 2 inches). Give your answer in terms of the nearest day.
In a certain region, monthly precipitation peaks at 24 inches in September and falls to a low of 4 inches in March. Identify the periods when the region is under flood conditions (greater than 22 inches) and drought conditions (less than 5 inches). Give your answer in terms of the nearest day.
Floods: July 24 through October 7. Droughts: February 4 through March 27
For the following exercises, find the amplitude, period, and frequency of the given function.
The displacement $\text{\hspace{0.17em}}h(t)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h(t)=8\mathrm{sin}(6\pi t),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.
The displacement $\text{\hspace{0.17em}}h(t)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h(t)=11\mathrm{sin}(12\pi t),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.
Amplitude: 11, period: $\text{\hspace{0.17em}}\frac{1}{6},\text{\hspace{0.17em}}$ frequency: 6 Hz
The displacement $\text{\hspace{0.17em}}h(t)\text{\hspace{0.17em}}$ in centimeters of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h(t)=4\mathrm{cos}\left(\frac{\pi}{2}t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.
For the following exercises, construct an equation that models the described behavior.
The displacement $\text{\hspace{0.17em}}h(t),$ in centimeters, of a mass suspended by a spring is modeled by the function $\text{\hspace{0.17em}}h(t)=\mathrm{-5}\text{\hspace{0.17em}}\mathrm{cos}\left(60\pi t\right),$ where $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is measured in seconds. Find the amplitude, period, and frequency of this displacement.
Amplitude: 5, period: $\text{\hspace{0.17em}}\frac{1}{30},$ frequency: 30 Hz
For the following exercises, construct an equation that models the described behavior.
A deer population oscillates 19 above and below average during the year, reaching the lowest value in January. The average population starts at 800 deer and increases by 160 each year. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$
A rabbit population oscillates 15 above and below average during the year, reaching the lowest value in January. The average population starts at 650 rabbits and increases by 110 each year. Find a function that models the population, $\text{\hspace{0.17em}}P,$ in terms of months since January, $\text{\hspace{0.17em}}t.$
$P(t)=-15\mathrm{cos}\left(\frac{\pi}{6}t\right)+650+\frac{55}{6}t$
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