Question

1. An airplane if flying horizontally at a constant height of 6 km above a fixed observation point. At a certain moment the angle of elevation θ is 30° and decreasing and the speed of the plane is 4 km/h. (a) How fast is 0 decreasing at this moment? (b) How fast is the distance between the plane and the observation point is changing at this moment? 2. Trajectory of a particle is described by parametrical equations as t,y P, where t ER A detector is placed at the point (-1,2). Find the coordinates of the particle when it will be at the closest distance from the detector 3. Use implicit differentiation to show that a function sin cosy 2y has a critical point whenever cosr = 0. Find the nature of these points. 4. Consider a triangle with sides α = 4 cm and b = 6 an. The angle θ is an angle between the sides a and b (a) Find the length of the third side of the triangle, c. Hint: use the law of cosines. (b) what is the rate of change of the third side c if the angle θ decreases at the rate of 0.5 rad/sec at the moment when θ--? 5. Consider a right angled triangle with base z cm and hypotenuse . 8cm. The base of the triangle changes with the rate 2cm/sec. Find the rate of change of the angle θ between the height and the hypotenuse of the triangle at the moment when Hint: What will be sin 0? 6. A temporary shelter is made from a piece of tarpaulin that is 6 m long and 4 m wide by folding it through the middle along the longer side and attaching to the ground at the corners (a) How far apart should the sides of the shelter be placed to maximize the enclosed volume? (b) Assume that the distance between the sides of the shelter decreases at the rate of 50 and to the trees with the rope that goes along the fold. Hint: Use the angle θ between the sides. cm/min. Find the rate of change of the enclosed volume at the moment when the 7. A particle moves in the anticlockwise direction along a circle with a radius of 13 em at the (a) Find the rate of change of the particle's coordinates (x,) with time at the moment (b) Challenge: Find the rate of change of the distance D between the particle and the 8. Consider a conical tank whose radius at the top is 4 cm and whose depth is 10 cm. It is distance is 2 m. Hint: Use law of cosines. constant rate of 10 revolutions per minute. when 12 cm and y 5cm Hint: express both r and y via angle of revolution, 6 point (13.0) at the moment when 1-12 cm and y-5 cm. being filled with water at the rate of 2 cubic em per minute. How fast is the water level rising when it is at depth 5 cm?

Answer 1

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