Let s be the set of all points that are a distance at

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Unformatted text preview: 1:46 AM PROBLEMS PLUS Page 883 1. Each edge of a cubical box has length 1 m. The box contains nine spherical balls with the same radius r . The center of one ball is at the center of the cube and it touches the other eight balls. Each of the other eight balls touches three sides of the box. Thus, the balls are tightly packed in the box. (See the figure.) Find r . (If you have trouble with this problem, read about the problem-solving strategy entitled Use analogy on page 58.) 2. Let B be a solid box with length L, width W, and height H . Let S be the set of all points that are a distance at most 1 from some point of B. Express the volume of S in terms of L, W, and H . 3. Let L be the line of intersection of the planes cx 1m y z c and x c y c z 1, where c is a real number. (a) Find symmetric equations for L. (b) As the number c varies, the line L sweeps out a surface S. Find an equation for the curve of intersection of S with the horizontal plane z t (the trace of S in the plane z t). (c) Find the volume of the solid bounded by S and the planes z 0 and z 1. 4. A plane is capable of flying at a speed of 180 km h in still air. The pilot takes off from an 1m 1m FIGURE FOR PROBLEM 1 airfield and heads due north according to the plane’s compass. After 30 minutes of flight time, the pilot notices that, due to the wind, the plane has actually traveled 80 km at an angle 5° east of north. (a) What is the wind velocity? (b) In what direction should the pilot have headed to reach the intended destination? 5. Suppose a block of mass m is placed on an inclined plane, as shown in the figure. The block’s N F W ¨ FIGURE FOR PROBLEM 5 descent down the plane is slowed by friction; if is not too large, friction will prevent the block from moving at all. The forces acting on the block are the weight W, where W mt ( t is the acceleration due to gravity); the normal force N (the normal component of the reactionary force of the plane on the block), where N n; and the force F due to friction, which acts parallel to the inclined plane, opposing the direction of motion. If the block is at rest and is increased, F must also increase until ultimately F reaches its maximum, beyond which the block begins to slide. At this angle s , it has been observed that F is proportional to n. Thus, when F is maximal, we can say that F s n, where s is called the coefficient of static friction and depends on the materials that are in contact. (a) Observe that N F W 0 and deduce that s tan s . (b) Suppose that, for s , an additional outside force H is applied to the block, horizontally from the left, and let H h. If h is small, the block may still slide down the plane; if h is large enough, the block will move up the plane. Let h min be the smallest value of h that allows the block to remain motionless (so that F is maximal). By choosing the coordinate axes so that F lies along the x-axis, resolve each force into components parallel and perpendicular to the inclined plane and show that h min sin (c) Show that m t cos n h min and m t tan h min cos s n m t sin s Does this equation seem reasonable? Does it make sense for l 90 ? s ? As Explain. (d) Let h max be the largest value of h that allows the block to remain motionless. (In which direction is F heading?) Show that h max m t tan s Does this equation seem reasonable? Explain. 883...
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