Related Rates of Change - The Burns Home Page

MCB4UI Worksheet 4.4Related Rates of Change1. Water flows into a rectangular pool whose dimensions are 12 m long, 8m wide, and 10 m deep. If water is entering the pool at the rate of 3cubic metres per second (hint: this is the rate of change in volume),how fast is the level of the water rising? ( hint: let x represent howdeep the water is at any specific time)2. A chemical cube is left out to dry, the drying process symmetricallycompacts the cube so that the volume decreases at a rate of 2 cubicmetres per minute.a) Find the rate of change of an edge of the cube when the volume is27 cubic metres.b) What is the rate of change of the surface of the cube at thispoint?3. In the bottom of an hourglass, a conical pile of sand is formed at therate of 12 cubic cm per minute. The radius of the base of the pile isalways equal to one-half its altitude. How fast is the altitude rising1 2when it is 6 cm deep? (note: volume of a cone is equal to3 π rh,wherer is the radius and h is the height of the cone)4. A math student is standing 30 metres from a straight section ofrailroad track. A train is approaching, moving along the track at 90kilometres per hour. How fast is the distance between the train andthe student decreasing when the train is 50 metres from the student?5. A coffee maker uses a filter in the shape of a cone, with the filterbeing 10 cm high and having a radius of 4cm. Coffee is flowing fromthe filter into a cup at a rate of 4 cm 3per second. At what rate isthe level of coffee in the filter falling when the coffee in thefilter is 4 cm deep?6. One end of a 13 metre ladder is on the ground, and the other end restson a vertical wall. If the bottom end is drawn away from the was at 3metres per second, how fast is the top of the ladder sliding down thewall when the bottom of the ladder is 5 metres from the wall?

7. Consider a variable right angle triangle ABC in a rectangularcoordinate system. Vertex A is the origin, the right angle is at7 2vertex B on the y axis, and vertex C is on the parabola y = x + 1. If4B starts at (0, 1) and moves upward at a constant rate of 2 units per7second. How fast is the area of the triangle increasing when t =2seconds?8. A balloon in the shape of a sphere is being inflated so that thevolume is increasing by 100 cubic centimetres per second. At what rateis the radius increasing when the radius is 9 cm?9. From the edge of a dock 4 metres above the surface of the water, arowboat is being hauled in by a rope and is approaching the base ofthe dock at the rate of 1 metre per second. How fast is the length ofrope changing when the boat is 2 metres from the dock?10. A baseball diamond is a 90 foot square. A ball is batted along thethird-base line at a constant speed of 100 feet per second. How fastis its distance from first-base changing when:a) It is halfway to third-base?b) It reaches third-base?11. Let A, D, C, and r be the area, diameter, circumference, and radius ofdr cma circle, respectively. At a certain instant, r=6 and = 3 . Finddt sthe rate of change of A with respect to:a) r b) D c) C d) t