"Related Rates hw"
MATH 111 WORKSHEET : RELATED RATES dx dy 1. Given y 2 5x 2 2 : x 1 : y 2 : 3 : find . dt dt dx dy 2. Given 8 y 3 x 2 3x : x 3 : y 1 : = 2 : find . dt dt dx dy 3. Given xy 5 x 2 y 3 8 : x 2 : y 3 : = 5 : find . dt dt 4. Let x be the length of a rectangle in inches and y the width of the rectangle in inches. If x is decreasing at a rate of 3 in/min and y is increasing at a rate of 4 in/min, then what is the rate of change of the area of the rectangle when the length is 10 inches and the width is 12 inches? Is the area increasing or decreasing at this time? 5. Let x be the length of the base of a right triangle in inches and y the height of the triangle in inches. If x is increasing at a rate of 2 in/min and y is decreasing at a rate of 3 in/min, then what is the rate of change of the area of the triangle when the base is 6 inches and the height is 8 inches? Is the area increasing or decreasing at this time? 6. A 25-foot ladder is placed against a building. The base of the ladder is slipping away from the building at a rate of 4 ft/min. Find the rate at which the top of the ladder is sliding down the building at the instant when the bottom of the ladder is 7 ft from the base of the building. 7. One car leaves a given point and travels north 30 miles per hour. Another car leaves the same point at the same time and travels west at 40 miles per hour. At what rate is the distance between the two cars changing at the instant when the cars have traveled 2 hours. 8. A rock is thrown into a still pond. The circular ripples move outward from the point of impact of the rock so that the radius of the area of ripples increases at a rate of 2 ft/min. Find the rate at which the area is changing at the instant the radius is 4 feet. 9. A spherical snowball is placed in the sun. The sun melts the snowball so that its radius 1 decreases inch per hour. Find the rate of change of the volume with respect to time at 4 the instant the radius is 4 inches. 10. A cube of ice melts at the rate of 15 cubic inches per hour. Find the rate of change of the length of an edge when an edge is 10 inches long. 11. Gas is being pumped into a spherical balloon at the rate of 5 cubic feet per minute. If the pressure is constant, find the rate at which the radius is changing when the diameter of 3 the balloon is feet. 2 12. A manufacturer of handcrafted wine racks has determined that the cost to produce x units per month is given by C .1x 2 10 ,000 . How fast is cost per month changing when production is increasing at the rate of 10 units per month and the production level is 100 units? 13. Given the revenue and cost functions R 50x .4 x 2 and C 5 x 15, where x is the daily production (and sales), find the following when 40 units are produced daily and the production is increasing 10 units per day. (a) The rate of change of revenue with respect to time (b) The rate of change of cost with respect to time. (c) The rate of change of profit with respect to time. 14. A balloon has a small leak and loses air at the constant rate of 3 ft 3 / min . How fast is the radius of the balloon decreasing when it is 20 feet long? 15. A particle is moving on the graph of the equation y x 4 2 x 3 5 x 2 85 . If the abscissa (x value) is increasing at the constant rate of 2 units per second, how fast is the ordinate (y value) increasing or decreasing, when x = 3? 16. The total cost C(q), in thousands of dollars, to manufacture q thousand radios is given by the equation C(q) = 4 + 21q - 2q 2 , provided 0 q 5. When the level of production is 3000 radios, production is increasing at the rate of 50 radios per month. Find the corresponding rate of change in cost. 17. A small balloon is released at a point 120 feet away from an observer who is at ground level. If the balloon goes straight up at a rate of 8 feet per second, how fast is the distance from the observer to the balloon increasing when the balloon is 50 feet off the ground? 18. Oil spilled from a ruptured tanker spreads in a circle whose area increases at a constant rate of 6 square miles per hour. How fast is the radius of the spill changing when the area is 9 square miles? 19. The area of a rectangle is decreasing at the rate of 5 inches2/minute. Find the rate of change of the length of the rectangle when the length is 10 inches and the width is 7 inches and the width is increasing 3 inches/minute. Some of these exercises are from Calculus With Applications by Lial, Greenwell and Miller and Calculus With Applications by Larson, Hostetler and Edwards.