MCB4UI Worksheet 4.4**Related** **Rates** **of** **Change**1. 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 **of**railroad 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 c**of**fee maker uses a filter in the shape **of** a cone, with the filterbeing 10 cm high and having a radius **of** 4cm. C**of**fee is flowing fromthe filter into a cup at a rate **of** 4 cm 3per second. At what rate isthe level **of** c**of**fee in the filter falling when the c**of**fee 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 **of**the dock at the rate **of** 1 metre per second. How fast is the length **of**rope 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 **of**dr 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