Ap calculus question

268 CHAPTER 3 Applications of Differentiation 66. Consider the situation in Exercise 51 if the cost of laying pipe under the river is considerably higher than the cost of laying pipe over land ($400,000/km). You may suspect that in some instances, the minimum distance possible under the river should be used, and P should be located 6 km from the refinery, directly across from the storage tanks. Show that this is never the case, no matter what the "under river" cost is 40 67. Consider the tangent line to the ellipse- 71. Let v, be the velocity of light in air and v2 the velocity of at a point (p, q) in the first quadrant. light in water. According to Fermat's Principle, a ray of light (a) Show that the tangent line has x-intercept at/p and will travel from a point A in the air to a point B in the water y-intercept b'/ q. by a path ACB that minimizes the time taken. Show that (b) Show that the portion of the tangent line cut off by the sin coordinate axes has minimum length a + b. (c) Show that the triangle formed by the tangent line and sin 02 the coordinate axes has minimum area ab. where 01 (the angle of incidence) and 02 (the angle of refrac- CAS 68. The frame for a kite is to be made from six pieces of wood. tion) are as shown. This equation is known as Snell's Law. The four exterior pieces have been cut with the lengths indicated in the figure. To maximize the area of the kite, how long should the diagonal pieces be? 72. Two vertical poles PQ and ST are secured by a rope PRS A 69. A point P needs to be located somewhere on the line AD going from the top of the first pole to a point R on the so that the total length L of cables linking P to the points ground between the poles and then to the top of the second A, B. and C is minimized (see the figure). Express L as a pole as in the figure. Show that the shortest length of such a function of x = | AP | and use the graphs of L and dL / dx to rope occurs when 01 = 02. estimate the minimum value of L. I 5 m 62 O 2m_1. 3 m D 73. The upper right-hand corner of a piece of paper, 12 in. by 70. The graph shows the fuel consumption c of a car (measured 8 in., as in the figure, is folded over to the bottom edge. How in gallons per hour) as a function of the speed v of the car. At very low speeds the engine runs inefficiently, so initially c would you fold it so as to minimize the length of the fold? In decreases as the speed increases. But at high speeds the fuel other words, how would you choose x to minimize y? consumption increases. You can see that c(v) is minimized for this car when v ~ 30 mi/h. However, for fuel efficiency. what must be minimized is not the consumption in gallons per hour but rather the fuel consumption in gallons per mile. Let's call this consumption G. Using the graph, estimate the speed at which G has its minimum value