3.6 Modeling with trigonometric equations  (Page 7/14)

$f(x)=-30\cos \left(\frac{x\pi }{6}\right)-20{\cos }^2\left(\frac{x\pi }{6}\right)+80[0,12]$

$f(x)=-18\cos \left(\frac{x\pi }{12}\right)-5\sin \left(\frac{x\pi }{12}\right)+100$ on the interval $[0,24]$

$f(x)=10-\sin \left(\frac{x\pi }{6}\right)+24\tan \left(\frac{x\pi }{240}\right)$ on the interval $[0,80]$

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.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of $105\text{°F}$ occurs at 5PM and the average temperature for the day is $85\text{°F}\text{.}$ 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 $84\text{°F}$ occurs at 6PM and the average temperature for the day is $70\text{°F}\text{.}$ Find the temperature, to the nearest degree, at 7AM.

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $47\text{°F}$ and $63\text{°F}$ during the day and the average daily temperature first occurs at 10 AM. How many hours after midnight does the temperature first reach $51\text{°F?}$

Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the temperature varies between $64\text{°F}$ and $86\text{°F}$ during the day and the average daily temperature first occurs at 12 AM. How many hours after midnight does the temperature first reach $70\text{°F?}$

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

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.