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chapter6Problems

# chapter6Problems - Chapter 6 Optimization 6.1 The sum of...

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Chapter 6 Optimization 6.1 The sum of two positive number is 20. Find the numbers (a) if their product is a maximum. (b) if the sum of their squares is a minimum. (c) if the product of the square of one and the cube of the other is a maximum. 6.2 A tram ride at Disney World departs from its starting place at t = 0 and travels to the end of its route and back. Its distance from the terminal at time t can be approximately described by the expression S ( t ) = 4 t 3 (10 - t ) (where t is in minutes, 0 < t < 10, and S is distance in meters.) (a) Find the velocity as a function of time. (b) When is the tram moving at the fastest rate? (c) At what time does it get to the furthest point away from its starting position? (d) Sketch the acceleration, the velocity, and the position of the tram on the same set of axes. 6.3 At 9 A.M. , car B is 25 km west of another car A . Car A then travels to the south at 30 km / h and car B travels east at 40 km / h. When will they be the closest to each other and what is this distance? v.2005.1 - September 4, 2009 1

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Math 102 Problems Chapter 6 6.4 A cannonball is shot vertically upwards from the ground with initial velocity v 0 = 15m / sec. It is determined that the height of the ball, y (in meters), as a function of the time, t (in sec) is given by y = v 0 t - 4 . 9 t 2 Determine the following: (a) The time at which the cannonball reaches its highest point, (b) The velocity and acceleration of the cannonball at t = 0 . 5 s, and t = 1 . 5 s. (c) The time at which the cannonball hits the ground. 6.5 (From Final Exam, Math 100, Dec 1997) A closed 3-dimensional box is to be constructed in such a way that its volume is 4500 cm 3 . It is also specified that the length of the base is 3 times the width of the base. Find the dimensions of the box which satisfy these conditions and have the minimum possible surface area. Justify your answer. 6.6 A box with a square base is to be made so that its diagonal has length 1. See Figure 6.1. (a) What height y would make the volume maximal? (b) What is the maximal volume?
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