MCV4U1 Worksheet 4.7. dh / dt if neither r nor h is constant?

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Esmond Lawson
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1 MCV4U1 Worksheet 4.7 This worksheet serves as an additional exercise to complement the lesson and the examples given. Worksheets may take more than one day to complete. If you are stuck, read again the notes taken in class, see the additional examples from the website, work with fellow students, come to the tutor lessons after school. Complete solutions to all questions are available on the website. When in doubt do more! 1. Area. The radius r and area A of a circle are related by the equation A r. Write an equation that relates da / dt to dr / dt.. Surface Area. The radius r and surface area S of a sphere are related by the equation S 4 r. Write an equation that relates ds / dt to dr / dt. 3. Volume. The radius r, height h, and volume V of a right circular cylinder are related by the equation V r h. a. How is dv / dt related to dh / dt if r is constant? b. How is dv / dt related to dr / dt if h is constant? c. How is dv / dt related to dr / dt and dh / dt if neither r nor h is constant? 4. Diagonals. If x, y, and z are lengths of the edges of a rectangular box, the common length of the box s diagonals is s x y z. How is ds / dt related to dx / dt, dy / dt, and dz / dt? 7. Air Traffic Control. An airplane is flying at an altitude of 7 mi and passes directly over a radar antenna as shown in the figure. When the plane is 10 mi from the antenna (s = 10), the radar detects that the distance s is changing at the rate of 300 mph. What is the speed of the airplane at that moment? 8. Sliding Ladder. A 13-ft ladder is leaning against a house (see figure) when its base starts to slide away. By the time the base is 1 ft from the house, the base is moving at the rate of 5 ft/sec. 5. Heating a Plate. When a circular plate of metal is heated in an oven, its radius increases at a rate of 0.01 cm/sec. At what rate is the plate s area increasing when the radius is 50 cm? 6. Changing Dimensions in a Rectangle. The length l of a rectangle is decreasing at the rate of cm/sec while the width w is increasing at the rate of cm/sec. When l = 1 cm and w = 5 cm, find the rates of change of a. the area, b. the perimeter, and c. the length of a diagonal of the rectangle. a. How fast is the top of the ladder sliding down the wall at that moment? b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing at that moment? c. At what rate is the angle between the ladder and the ground changing at that moment?

2 9. Growing Sand Pile. Sand falls from a conveyor belt at the rate of 10 m 3 /min onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How hast are the (a) height and (b) radius changing when the pile is 4 m high? Give your answer is cm/min. 10. Draining Conical Reservoir. Water is flowing at the rate of 50 m 3 /min from a concrete conical reservoir (vertex down) of base radius 45 m and height 6 m. (a) How fast is the water level falling when the water is 5 m deep? (b) How fast is the radius of the water s surface changing at that moment? Give your answer in cm/min. 14. Rising Balloon. A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance between the bicycle and balloon increasing 3 sec later (see figure)? 11. Growing Raindrop. Suppose that a droplet of mist is a perfect sphere and that, through condensation, the droplet picks up moisture at a rate proportional to its surface area. Show that under these circumstances the droplet s radius increases at a constant rate. 1. Inflating Balloon. A spherical balloon is inflated with helium at the rate of 100 ft 3 /min. a. How fast is the balloon s radius increasing at the instant the radius is 5 ft? b. How fast is the surface area increasing at that instant? 13. Hauling in a Dinghy. A dinghy is pulled toward a dock by a rope from the bow through a ring on the dock 6 ft above the bow as shown in the figure. The rope is hauled in at the rate of ft/sec. 15. Cost, Revenue, and Profit. A company can manufacture x items at a cost of c(x) dollars, a sales revenue of r(x) dollars, and a profit of p x r x c x dollars (all amounts in thousands). Find dc / dt, dr / dt, and dp / dt for the following values of x and dx / dt. 3 x 9x, c x x 6x 15x, r and dx / dt 0.1 when x. a. How fast is the boat approaching the dock when 10 ft of rope are out? b. At what rate is angle changing at that moment?

3 16. Making Coffee. Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in 3 /min. 19. Building s Shadow. On a morning of a day when the sun will pass directly overhead, the shadow of an 80-ft building on level ground is 60 ft long as shown in the figure. At the moment in question, the angle the sun makes with the ground is increasing at the rate of 0.7 degrees/min. At what rate is the shadow length decreasing? Express your answer in in/.min, to the nearest tenth. (Remember to use radians.) a. How fast is the level in the pot rising when the coffee in the cone is 5 in. deep? b. How fast is the level in the cone falling at that moment? 17. Moving Shadow. A man 6 ft tall walks at the rate of 5 ft/sec toward a streetlight that is 16 ft above the ground. At what rate is the length of his shadow changing when he is 10 ft from the base of the light? 0. A particle is moving along the curve 10 y. If dx / dt 3 cm/sec, find 1 x dy / dt at the point where x. 18. Speed Trap. A highway patrol airplane flies 3 mi above a level, straight road at a constant rate of 10 mph. The pilot sees an oncoming car and with radar determines that at the instant the line-of-sight distance from plane to car is 5 mi the line-of-sight distance is decreasing at the rate of 160 mph. Find the car s speed along the highway.

4 MCV4U1 Worksheet 4.7 Solutions

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Math 241 Homework 6 Solutions

Math 241 Homework 6 Solutions Math 241 Homework 6 s Section 3.7 (Pages 161-163) Problem 2. Suppose that the radius r and surface area S = 4πr 2 of a sphere are differentiable functions of t. Write an equation that relates ds/ to /.