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IIT MATH 149 - MATH 149 Recitation 4 Problems

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MATH 149 Spring 2010 Recitation 4 Problems Wednesday, 17 March 20101. A poster of 500 square inches is to have a margin of 6 inches at the top and 4 inches at each side andthe bottom. What dimensions yield the largest printed area?2. An athletic field with a 400 perimeter consists of a rectangle with a semicircle at each end. Find thedimensions of the field so that the area of the rectangular portion is the largest possible.3. Find the tangent line to the curve y D 4  x2at a point in the first quadrant that cuts from the firstquadrant a triangle of minimum area.4. Describe the isosceles triangle of maximum area if two sides have a fixed length s.5. An object with weight W is dragged along a horizontal plane by a force acting along a rope attachedto the object. If the rope makes an angle  with the plane, then the magnitude of the force isF D W sin  C cos where  is a positive constant called the coefficient of friction and where 0    =2. Show thatF is minimized when tan D .6. The frame for a kite is to be made from six pieces of wood. The four exterior pieces have been cutwith the lengths indicated in the figure, two of length a and two of length b. To maximize the areaof the kite, how long should the diagonal pieces be? (See figure of kite on page 285 in Stewart.)7. A boat leaves a dock at 2:00P.M. and travels due south at a speed of 20 km/h. Another boat hasbeen heading due east at 15 km/h and reaches the same dock at 3:00P.M. At what time were thetwo boats closest together?8. Show that for motion in a straight line with constant acceleration a, initial velocity v0, and initialdisplacement s0, the displacement after time t iss D12at2C v0t C s0:9. An automobile is traveling on a straight road at 88 feet per second (60 mph). What constant (nega-tive) acceleration is required to stop the car in 40 feet?10. A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds isa.t/ D 60t, at which time the fuel is exhausted and it becomes a freely “falling” body. Fourteenseconds later, the rocket’s parachute opens, and the (downward) velocity slows linearly to 18 ft/sin 5 s. The rocket then “floats” to the ground at that rate.(a) Determine the position function s and the velocity function v (for all times t). Sketch the graphsof s and v.(b) At what time does the rocket reach its maximum height, and what is that height?(c) At what time does the rocket land?1MATH 149 Spring 2010 Recitation 4 Problems Wednesday, 17 March 201011. A canister is dropped from a helicopter 500 meters above the ground. Its parachute does not open,but the canister has been designed to withstand an impact velocity of 100 meters per second. Will itburst?12. In an automobile race along a straight road, car A passed car B twice. Prove that at some timeduring the race their accelerations were equal. State the assumptions that you


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IIT MATH 149 - MATH 149 Recitation 4 Problems

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