Related Rates. Instant (true at an instant)

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1 Related Rates Name Related Rates Day 1: 1. Assume that oil spilled from a ruptured tanker spreads in a circular pattern whose area increases at a constant rate of 5 m 2 /s. How fast is the radius of the spill increasing when the radius of the spill is 25 m? A = πr 2 2. Sheila walks to the Dead Sea and throws a rock into the lake. Since the Sea is calm, ripples in the shape of concentric circles are formed on the water. If the radius of the outer ripple is increasing at a rate of 2 feet per second, at what rate is the total area of disturbed water changing when the radius is 5 feet? A = πr 2 3. A television camera at ground level is filming the lift-off of a space shuttle. The shuttle is rising vertically according to the position equation s = 50t 2, where s is measured in feet and t is measured in seconds. The camera is 2000 feet from the launch pad. Find the rate of change in the angle of elevation of the camera at 10 seconds after lift-off. (Hint: use a trig function)

2 4. A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1 ft/s, how fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 ft from the wall? x 2 + y 2 = z 2 5. A railroad track and a road cross at right angles. An observer stands on the road 70 meters south of the crossing and watches an eastbound train traveling at 60 meters per second. At what rate is the distance between the observer and the train changing 4 seconds after the train passes through the intersection? x 2 + y 2 = z 2

3 Related Rates Day 2: For each problem, set up and solve. Show all of your work. Include units in your answers. 1. Cylinder (recognize that r is actually a constant) A cylinder filled with water has a 3 foot radius and a 10 foot height. It is drained such that the depth of the water is decreasing at 0.1 feet per second. How fast is the water draining from the tank? 2. Cone filling/draining A water tank in the shape of a right circular cone has a height of 14 feet. The top rim of the tank is a circle with a radius of 5 feet. The depth of the water is decreasing at 1 foot per minute. What is the rate of change of the volume when the depth of the water is 6 feet? 3. Cone Corn is poured through a chute at a rate of 10 cubic feet per minute and falls in a conical pile whose bottom radius is always half the height. How fast will the radius of the base change when the pile is 8 feet high?

4 4. Shadow A light is on the top of a 12 foot tall pole. A 5 foot 6 inch tall person is walking away from the pole at a rate of 2 feet/second. At what rate is the tip of the shadow moving away from the pole when the person is 25 feet from the pole? 5. A trough is 15 ft long and 4 ft across the top, as shown in the figure. Its ends are isosceles triangles with height 3 ft. Water runs into the trough at the rate of 2.5 ft 3 /min. How fast is the water level rising when it is 2 ft deep? How does the answer change if the trough is wider or narrower across the top?

5 Related Rates Day 3: For each problem, draw a diagram, set up and solve, showing all of your work. Include units in your answers. 1. A particle moves along the curve y 3 1 x. As it reaches the point (2,3) the y-coordinate is increasing at a rate of 4 cm/sec. How fast is the x-coordinate of the point changing at this instant? 2. Cylinder Oil Slick (AB 2008 #3) 3. Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 10 in 3 /min. The cone is 6 in high and has a radius of 6 in. The coffeepot has a radius of 6 in. 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?

6 4. A person whose height is 6 feet is walking away from the base of a streetlight along a straight path at a rate of 4 feet per second. If the height of the streetlight is 15 feet, what is the rate at which the person s shadow is lengthening? 5. An object is moving along the graph of y = x 2. When it reaches the point (2,4) the x-coordinate of the object is decreasing at the rate of 3 units per second. Give the rate of change of the distance between the objet and the point (0,1) at the instant when the object is at (2,4).