Exercise 15.5. In the baton example, notice that the tip of the baton makes a downward loop at around t = 0.5 sec. Find the time (to four decimal places) when the tip of the baton is at the bottom of the loop, and find its height at that moment.

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Exercise 15.6. In the baton example, suppose that there is a wall located at exactly x = 10 ft. Use Newton's method to find (to three decimal places) when the tip of the baton hits the wall.
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Exercise 15.7 . You are riding a double ferris wheel at an amusement park. The large circle has radius 40 ft, and the smaller circle has radius 10 ft. (See the diagram below.) Your seat rotates clockwise around the smaller circle once every 6 sec, and the center of the small circle rotates counterclockwise around the big circle once every 20 sec. At time t = 0 you are at the lowest possible point (50 ft below the center of the large circle). Call that point the origin. First write parametric equations for your position at time t. Then use Newton's method to find (to at least three places past the decimal point) the instant when you reach the highest point during the first revolution of the large circle. Also find your height at that moment.

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Exercise 15.8. Two mine shafts meet at an angle of , as shown in the diagram. The horizontal shaft has a width of 7 ft, while the entrance shaft is 9 ft wide. What is the longest ladder that can negotiate the turn? Neglect the thickness of the ladder, and assume it is not tipped as it is maneuvered around the corner. At a given moment the ladder makes an angle of as shown. First convince yourself that the problem can be rephrased as follows: Find the angle at which the length l = AB is minimal, and then find the corresponding value of l. To do this, first express l = AC + CB in terms of (notice that can be expressed in terms of ). Then set dl/d = 0, and use Newton's method with = to find and l to at least 3 significant figures of accuracy.

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