91Ó°ÊÓ

Temperature models: The average temperature on Valentine's Day in Sydney, Australia, can be modeled with the equation \(T(t)=4 \cos \left(\frac{\pi}{12} t+\frac{9 \pi}{12}\right)+22\), where \(T\) is the temperature in Celsius and \(t\) is the time of day ( \(t=0\) corresponds to midnight). Use the model to (a) find the period of the model; (b) find the average minimum and maximum temperature; and (c) when these "extreme" temperatures occur.

Temperature models: The average temperature on Valentine's Day in Sydney, North Dakota, can be modeled with the equation \(T(t)=5 \cos \left(\frac{\pi}{12} t+\frac{9 \pi}{12}\right)-8\), where \(T\) is the temperature in Celsius and \(t\) is the time of day ( \(t=0\) corresponds to midnight). Use the model to (a) find the period of the model; (b) find the average minimum and maximum temperature; and (c) when these extreme temperatures occur.

Wave height: A data buoy placed off the coast of Santa Cruz, California, measures wave height and transmits the information to a monitoring station. For the minute 12:28 PDT (low tide), the wave height can be modeled with the equation \(y=2.6 \sin \left(\frac{2 \pi}{6} t\right)-0.6\), where \(t\) is measured in seconds and \(y\) is in feet \((y=0\) corresponds to the height of calm sea between high and low tide). Use the model to find (a) the time between each wave and (b) wave height from peak to trough.

A heavy wind is kicking up ocean swells approximately \(10 \mathrm{ft}\) high (from crest to trough), with wavelengths of \(250 \mathrm{ft}\). (a) Find an equation that models these swells. (b) Graph the equation. (c) Determine the height of a wave measured \(200 \mathrm{ft}\) from the trough of the previous wave.

In what quadrant does the angle \(t=3.7\) terminate? What is the reference arc?

Circumscribed polygons: The perimeter of a regular polygon circumscribed about a circle of radius \(r\) is given by \(P=2 n r \tan \left(\frac{\pi}{n}\right)\), where \(n\) is the number of sides \((n \geq 3)\) and \(r\) is the radius of the circle. Given \(r=10 \mathrm{~cm}\), (a) What is the circumference of the circle? (b) What is the perimeter of the polygon when \(n=4\) ? Why? (c) Calculate the perimeter of the polygon for \(n=10,20,30\), and 100 . What do you notice? See Exercise 53 from Section 4.1.

The area of a regular polygon circumscribed about a circle of radius \(r\) is given by \(A=n r^{2} \tan \left(\frac{\pi}{n}\right)\), where \(n\) is the number of sides \((n \geq 3)\) and \(r\) is the radius of the circle. Given \(r=10 \mathrm{~cm}\), a. What is the area of the circle? b. What is the area of the polygon when \(n=4\) ? Why? c. Calculate the area of the polygon for \(n=10,20,30\), and 100 . What do you notice?

The perimeter of a regular polygon circumscribed about a circle of radius \(r\) is given by the formula \(P=2 n r \sin \left(\frac{\pi}{n}\right) \sec \left(\frac{\pi}{n}\right)\), where \(n\) represents the number of sides. (a) Verify the formula for a square circumscribed about a circle with radius \(4 \mathrm{~m}\). (b) Find the perimeter of a dodecagon (12 sides) circumscribed about the same circle.

$$ \text { Use a reference arc to determine the value of } \cos \left(\frac{29 \pi}{6}\right) \text {. } $$

Given \(\sin 212^{\circ} \approx-0.53\), find another angle \(\theta\) in \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfies \(\sin \theta \approx-0.53\) without using a calculator.