AP Calculus AB/EC Related Rates Draw a gure for each problem. Label all variables. Identify an equation that relates the rates given. Differentiate with respect to time. Answer the question in a complete sentence. Be sure to include units of measure. CIRCLEXSFHERE l. 2. An oil tank spills oil that spreads in a circular pattern whose radius increases at the rate of 50 feet/min. How fast are both the circumference and area of the spill increasing when the radius of the spill is 20 feet? The radius of a circle is decreasing at a constant rate of 11.] centimeters per second. In terms of the circumference C. what is the rate of change of the area of the circle. in square centimeters per second? . A balloon is being inated by pumped air at a rate of 2 cubic inches per second. How fast is the diameter of the balloon increasing when the radius is \"/2 inch? (volume of d . a sphere is l' = 31w" and surface area of a sphere is A = 4m") RECTAN GLE 4. The length of a rectangle is increasing at 3 ft [min and the width is decreasing at 2 ftfmin. When the length is 50 : and the width is 20 it. what is the rate at which the area is changing? TRIANGLE 5. A 15-foot ladder is leaning against a wall. The foot of the ladder is being pulled out at a rate of 3 inches per second. How fast is the top of the ladder falling when it is exactlyr 12 feet from the ground? Nancy Nerd is jogging westward along EZ Street, running toward the intersection with Fast Lane at 14 feet per second. Jerry Atrick is jogging toward that same intersection, going north on Fast Lane at 12 feet per second. Nancy is .300 feet from the intersection and Jerry is 160 feet from the intersection. How fast is the distance between the two runners decreasing? . Dorothg,r who is standing at point A, watches the Professor in the balloon rise straight up from point C. Dorothy is 80 meters west of point C. The balloon is rising at a constant rate of 5 meters per second. Find the rate of change of B when the balloon is 6'0 meters off the ground. . a ladder 1t) feet long is leaning against a building so that one end is on level ground and the other end is on the wall as shown in the gure. The bottom of the ladder is moved away from the wall at the constant rate of '5': foot per second. Find the rate of change in square feet per second of the area of triangle formed by the ladder and the wall when the bottom of the ladder is 3 feet from the wall. SIMILAR TRIANGLES 9. A light is on the top ofa 12 it pole and a 5 ft 6 in person is walking away from the pole at a rate oi'2 itfsec. At what rate is the tip oi'the shadow moving away from the pole when the person is 25 it from the pole? it}. A spotlight is on the ground 20 it away from a wall and a 6 ft tall person is walking towards the wall at a rate oi'2.5 i'thee. How fast is the height ofthe shadow changing when the Person is 8 feet from the wall? CYLINDER l 1.3%, cylinder is changing such that its radius is decreasing by 0.5 inches per second and its height is increasing by 1 inch per second. At what rate is the volume changing when the height is 8 inches and the radius is 3 inches? {Volume ofa cylindrical tank is V = :rrlii and surface area is A = EmulHZm-J ] 12A cylindrical tank of 10 ft is being lled with wheat at the rate of314 cubic feet per minute. How fast is the depth of the wheat increasing? (Volume ot'a cylindrical tank is V = will: and surface area is A = 2.=i'rii+2irrI ] CONE 13.8uppose you are drinking root beer from a conical paper cup. The cup has a diameter ofB cm and a depth 01' 10 cm. ts you suck on the straw, the root beer leaves the cup at the rate oi\"? em' isce. At what rates is the level of the liquid in the cup changing when the liquid is 6 cm deep? [Volume of a cone is l\" = Earl}: and surface area is A = rrri' + in" :- l4.The volume V of a cone is increasing at the rate of cubic inches per second. At the instant when the radius r of the cone is 3 in, its volume is cubic inches and the radius is increasing at V2 inches per second. [Volume oi'a cone is V = Emil: and surface area is A = rm" + arr: i a. At the instant when the radius oi'the cone is 3 inches. what is the rate oi'ehange in the area of its base? 1:. At the instant when the radius of the cone is 3 inches. what is the rate of change in the height h? [SA snow cone cup is an inverted right circular cone. The cup has a height oft\": inches and a radius of 3 inches. Snow cone liquid is leaking out of the cup and down your arm at a constant rate of a cubic inches per minutes. How fast is the radius of the liquid falling when the depth of the liquid is 2 inches? How fast is the radius of the liquid decreasing when the radius of the liquid is 2 inches? [Volume of a cone is V = liar\": and surface area is A = mf+2tr=l