91Ó°ÊÓ

In Exercises \(1-20\), plot the graph of the polar equation by hand. Carefully label your graphs. Circle: \(r=6 \sin (\theta)\)

The London Eye is a popular tourist attraction in London, England and is one of the largest Ferris Wheels in the world. It has a diameter of 135 meters and makes one revolution (counterclockwise) every 30 minutes. It is constructed so that the lowest part of the Eye reaches ground level, enabling passengers to simply walk on to, and off of, the ride. Find a sinsuoid which models the height \(h\) of the passenger above the ground in meters \(t\) minutes after they board the Eye at ground level.

Plot the point given in polar coordinates and then give three different expressions for the point such that (a) \(r<0\) and \(0 \leq \theta \leq 2 \pi\) (b) \(r>0\) and \(\theta \leq 0\) (c) \(r>0\) and \(\theta \geq 2 \pi\) $$ \left(\frac{1}{3}, \frac{3 \pi}{2}\right) $$

The table below lists the average temperature of Lake Erie as measured in Cleveland, Ohio on the first of the month for each month during the years \(1971-2000 .^{19}\) For example, \(t=3\) represents the average of the temperatures recorded for Lake Erie on every March 1 for the years 1971 through 2000 . $$ \begin{array}{|l|r|r|r|r|r|r|r|r|r|r|r|r|} \hline \text { Month } & & & & & & & & & & & & \\ \text { Number, } t & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\ \hline \begin{array}{l} \text { Temperature } \\ \left({ }^{\circ} \mathrm{F}\right), T \end{array} & 36 & 33 & 34 & 38 & 47 & 57 & 67 & 74 & 73 & 67 & 56 & 46 \\ \hline \end{array} $$ (a) Using the techniques discussed in Example 11.1.2, fit a sinusoid to these data. (b) Using a graphing utility, graph your model along with the data set to judge the reasonableness of the fit. (c) Use the model you found in part 8 a to predict the average temperature recorded for Lake Erie on April \(15^{\text {th }}\) and September \(15^{\text {th }}\) during the years \(1971-2000 .^{20}\) (d) Compare your results to those obtained using a graphing utility.

Use the Law of Cosines to find the remaining side(s) and angle(s) if possible. $$ a=5, b=5, c=5 $$

The HMS Sasquatch leaves port on a bearing of \(\mathrm{N} 23^{\circ} \mathrm{E}\) and travels for 5 miles. It then changes course and follows a heading of \(\mathrm{S} 41^{\circ} \mathrm{E}\) for 2 miles. How far is it from port? Round your answer to the nearest hundredth of a mile. What is its bearing to port? Round your angle to the nearest degree.

From a point 300 feet above level ground in a firetower, a ranger spots two fires in the Yeti National Forest. The angle of depression \(^{7}\) made by the line of sight from the ranger to the first fire is \(2.5^{\circ}\) and the angle of depression made by line of sight from the ranger to the second fire is \(1.3^{\circ}\). The angle formed by the two lines of sight is \(117^{\circ}\). Find the distance between the two fires. Round your answer to the nearest foot. (Hint: In order to use the \(117^{\circ}\) angle between the lines of sight, you will first need to use right angle Trigonometry to find the lengths of the lines of sight. This will give you a Side-Angle-Side case in which to apply the Law of Cosines.)

The Colonel spots a campfire at a of bearing \(\mathrm{N} 42^{\circ} \mathrm{E}\) from his current position. Sarge, who is positioned 3000 feet due east of the Colonel, reckons the bearing to the fire to be \(\mathrm{N} 20^{\circ} \mathrm{W}\) from his current position. Determine the distance from the campfire to each man, rounded to the nearest foot.

Approximate the component form of the vector \(\vec{v}\) using the information given about its magnitude and direction. Round your approximations to two decimal places. \(\|\vec{v}\|=63.92\); when drawn in standard position \(\vec{v}\) makes a \(78.3^{\circ}\) angle with the positive \(x\) -axis

The captain of the SS Bigfoot sees a signal flare at a bearing of \(\mathrm{N} 15^{\circ} \mathrm{E}\) from her current location. From his position, the captain of the HMS Sasquatch finds the signal flare to be at a bearing of \(\mathrm{N} 75^{\circ} \mathrm{W}\). If the SS Bigfoot is 5 miles from the HMS Sasquatch and the bearing from the SS Bigfoot to the HMS Sasquatch is \(\mathrm{N} 50^{\circ} \mathrm{E}\), find the distances from the flare to each vessel, rounded to the nearest tenth of a mile.