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Chapter 1: Problem 66
Convert the following expressions to the indicated base. \(\log _{2}\left(x^{2}+1\right)\) using base \(e\)
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Kelly has finished a picnic on an island that is \(200 \mathrm{m}\) off shore (see figure). She wants to return to a beach house that is \(600 \mathrm{m}\) from the point \(P\) on the shore closest to the island. She plans to row a boat to a point on shore \(x\) meters from \(P\) and then jog along the (straight) shore to the house. a. Let \(d(x)\) be the total length of her trip as a function of \(x .\) Graph this function. b. Suppose that Kelly can row at \(2 \mathrm{m} / \mathrm{s}\) and jog at \(4 \mathrm{m} / \mathrm{s}\). Let \(T(x)\) be the total time for her trip as a function of \(x\). Graph \(y=T(x)\) c. Based on your graph in part (b), estimate the point on the shore at which Kelly should land in order to minimize the total time of her trip. What is that minimum time?
Assume that \(b > 0\) and \(b \neq 1\). Show that \(\log _{1 / b} x=-\log _{b} x\).
Verify that the function $$ D(t)=2.8 \sin \left(\frac{2 \pi}{365}(t-81)\right)+12 $$ has the following properties, where \(t\) is measured in days and \(D\) is measured in hours. a. It has a period of 365 days. b. Its maximum and minimum values are 14.8 and \(9.2,\) respectively, which occur approximately at \(t=172\) and \(t=355\) respectively (corresponding to the solstices). c. \(\overline{D(81)}=12\) and \(D(264)=12\) (corresponding to the equinoxes).
The shortest day of the year occurs on the winter solstice (near December 21) and the longest day of the year occurs on the summer solstice (near June 21 ). However, the latest sunrise and the earliest sunset do not occur on the winter solstice, and the earliest sunrise and the latest sunset do not occur on the summer solstice. At latitude \(40^{\circ}\) north, the latest sunrise occurs on January 4 at 7: 25 a.m. ( 14 days after the solstice), and the earliest sunset occurs on December 7 at 4: 37 p.m. ( 14 days before the solstice). Similarly, the earliest sunrise occurs on July 2 at 4: 30 a.m. (14 days after the solstice) and the latest sunset occurs on June 7 at 7: 32 p.m. ( 14 days before the solstice). Using sine functions, devise a function \(s(t)\) that gives the time of sunrise \(t\) days after January 1 and a function \(S(t)\) that gives the time of sunset \(t\) days after January \(1 .\) Assume that \(s\) and \(S\) are measured in minutes and \(s=0\) and \(S=0\) correspond to 4: 00 a.m. Graph the functions. Then graph the length of the day function \(D(t)=S(t)-s(t)\) and show that the longest and shortest days occur on the solstices.
Design a sine function with the given properties. It has a period of 24 hr with a minimum value of 10 at \(t=3\) hr and a maximum value of 16 at \(t=15 \mathrm{hr}\)
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