MATH
Group HW 9

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4.5 #34: To get the best view of the Statue of Liberty in Figure 4.76, you should be at the position where theta is a maximum. If the statue stands 92 meters high, including the pedestal, which is 46 meters high, how far from the base should you be? [Hint: Find a formula for theta in terms of your distance from the base. Use this function to maximize theta, noting that 0< theta< pi/2.] From Figure 4.76, we can create a free body diagram which includes only the triangle with the sides and angles in question. Since we are looking for the distance from the base using the distance from the top of the pedestal, we will actually have two triangles. A diagram can be seen below in Figure 1. x Figure 1: Diagram of the angles and distances from the Statue of Liberty Looking at the entire triangle (the two smaller triangles put together), we have right angle and can use tangent to express theta. We will call the angle of the other smaller triangle, y. Looking at the smaller triangle, we can see that tan(y)=46/x. Then we can look at the triangle as a whole again and see that the entire angle is ( +y). Therefore, tan( +y)=92/x. ө ө We know that +y-y= . Using this stupid statement we can make a mathematical ө ө marvel by adding parentheses. Thus, ( +y)-y= . Genius! We know that +y=arctan(92/ ө ө ө x). We also know that y=arctan(46/x). So, we can put these two equations together forming: ( +y)-y= arctan(92/x)-arctan(46/x)= . Since we want to maximize theta, we ө ө can take the derivative of this equation and solve for x (the distance from the base) [not so much fun so please see Appendix A for the rest of the problem]. ө 46 92 46 {

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4.6 # 24: A train is traveling at .8 km/min along a long, straight track, moving in the direction shown in the figure. A movie camera, .5 km away from the track, is focused on the train. a) Express z , the distance between the camera and the train, as a function of x. We can express z(x) using the Pythagorean Theorem since the camera and the track form a right angle.
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