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$x$ | $y$ |
$0$ | $5$ |
$1$ | $-3$ |
$2$ | $5$ |
$3$ | $13$ |
$4$ | $5$ |
$5$ | $-3$ |
$6$ | $5$ |
$5-8\mathrm{sin}\left(\frac{x\pi}{2}\right)$
$x$ | $y$ |
$-3$ | $-1-\sqrt{2}$ |
$-2$ | $-1$ |
$-1$ | $1-\sqrt{2}$ |
$0$ | $0$ |
$1$ | $\sqrt{2}-1$ |
$2$ | $1$ |
$3$ | $\sqrt{2}+1$ |
$x$ | $y$ |
$-1$ | $\sqrt{3}-2$ |
$0$ | $0$ |
$1$ | $2-\sqrt{3}$ |
$2$ | $\frac{\sqrt{3}}{3}$ |
$3$ | $1$ |
$4$ | $\sqrt{3}$ |
$5$ | $2+\sqrt{3}$ |
$\mathrm{tan}\left(\frac{x\pi}{12}\right)$
For the following exercises, graph the given function, and then find a possible physical process that the equation could model.
$f(x)=-30\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{x\pi}{6}\right)-20\text{\hspace{0.17em}}{\mathrm{cos}}^{2}\left(\frac{x\pi}{6}\right)+80\text{\hspace{0.17em}}\text{\hspace{0.17em}}[0,12]$
$f(x)=-18\text{\hspace{0.17em}}\mathrm{cos}\left(\frac{x\pi}{12}\right)-5\text{\hspace{0.17em}}\mathrm{sin}\left(\frac{x\pi}{12}\right)+100\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}[0,24]$
Answers will vary. Sample answer: This function could model temperature changes over the course of one very hot day in Phoenix, Arizona.
$f(x)=10-\mathrm{sin}\left(\frac{x\pi}{6}\right)+24\text{\hspace{0.17em}}\mathrm{tan}\left(\frac{x\pi}{240}\right)\text{\hspace{0.17em}}$ on the interval $\text{\hspace{0.17em}}[0,80]$
For the following exercise, construct a function modeling behavior and use a calculator to find desired results.
A city’s average yearly rainfall is currently 20 inches and varies seasonally by 5 inches. Due to unforeseen circumstances, rainfall appears to be decreasing by 15% each year. How many years from now would we expect rainfall to initially reach 0 inches? Note, the model is invalid once it predicts negative rainfall, so choose the first point at which it goes below 0.
9 years from now
For the following exercises, construct a sinusoidal function with the provided information, and then solve the equation for the requested values.
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $\text{\hspace{0.17em}}105\text{\xb0F}\text{\hspace{0.17em}}$ occurs at 5PM and the average temperature for the day is $\text{\hspace{0.17em}}85\text{\xb0F}\text{.}\text{\hspace{0.17em}}$ Find the temperature, to the nearest degree, at 9AM.
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $\text{\hspace{0.17em}}84\text{\xb0F}\text{\hspace{0.17em}}$ occurs at 6PM and the average temperature for the day is $\text{\hspace{0.17em}}70\text{\xb0F}\text{.}\text{\hspace{0.17em}}$ Find the temperature, to the nearest degree, at 7AM.
$56\text{}\xb0\text{F}$
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $\text{\hspace{0.17em}}47\text{\xb0F}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}63\text{\xb0F}\text{\hspace{0.17em}}$ during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach $\text{\hspace{0.17em}}51\text{\xb0F?}$
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $\text{\hspace{0.17em}}64\text{\xb0F}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}86\text{\xb0F}\text{\hspace{0.17em}}$ during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach $\text{\hspace{0.17em}}70\text{\xb0F?}$
$\text{\hspace{0.17em}}1.8024\text{\hspace{0.17em}}$ hours
A Ferris wheel is 20 meters in diameter and boarded from a platform that is 2 meters above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 6 minutes. How much of the ride, in minutes and seconds, is spent higher than 13 meters above the ground?
A Ferris wheel is 45 meters in diameter and boarded from a platform that is 1 meter above the ground. The six o’clock position on the Ferris wheel is level with the loading platform. The wheel completes 1 full revolution in 10 minutes. How many minutes of the ride are spent higher than 27 meters above the ground? Round to the nearest second
4:30
The sea ice area around the North Pole fluctuates between about 6 million square kilometers on September 1 to 14 million square kilometers on March 1. Assuming a sinusoidal fluctuation, when are there less than 9 million square kilometers of sea ice? Give your answer as a range of dates, to the nearest day.
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