Questions:

3. (8 points) Uri is lling a cone with molten aluminum. The cone is inside-down, so the \"base\" is at the top of the cone and the vertex at the bottom. as shown in the diagram. The base is a circular disk with radius 7 cm and the height of the cone is 12 cm. Recall . 1 . . . that the volume of a cone 1s 52411. where A is the area of the base and h. is the height of the cone (i.e., the vertical distance from the vertex to the base}. (Note that the diagram may not be to scale.) 7 cm 12 cm molten alulnuuun (a) (3 points) Write a. formula in terms of h. for the volume V of molten aluminum, in cm3. in the cone if the molten aluminum in the cone reaches a height of h. cm. (1)) (5 points) The height of molten aluminum is rising at 3 cmfsec at the moment when the molten aluminum in the cone has reached a height of 11 em. What is the rate, in cud/see, at which Uri is pouring molten alnmimnu into the (tone at that moment? 4. (8 points} Octavius1 the giant octopus, escaped from a storage tank at the zoo near the harbor. He managed to get into the bay. but the coast guard could keep track of him with a homing device that had been attached to Octavius at the zoo. (a) (4 points) The coast guard station is 2 kilometers (2000 meters) down the beach from where Octavius entered the bay. If the octopus was moving directly away from the shore at a constant rate of 25 meters per minute. how fast was the distance between the coast guard and Octavius changing when the octopus was 2000 meters from the shore? (b) (4 points) At what rate is the angle formed by the beach and the line that gives distance from the coast guard station to Octavius changing when Octavius is 2000 meters from the shore? Example. A fisherman hooks a trout and reels in his line at 4 cm per second. Assume the tip of the fishing rod is 3 meters above the water and directly above the angler, and the fish tip of is pulled horizontally directly toward the angler. Pole 1. Find the horizontal speed of the trout when it is 5 meters from the fisherman. line know : dc - - 4 Cm /S X I'm 10 0 CM = - 0.04 m/s at a RC a = 3 m 3 m b = 5 m Find : db fish dt water a2 + 62 = C2 32 +62 = c2 32 +52 = C2 0+ pb db = z c dc 9+25 = C2 5 db = 834 (-0.04) C2 = 34 C = V 34 db ( 34 ( 0 .04 ) ~ 5 2. Find the rate of change of the angle made between the water and the fishing line when SOM the trout is 5 meters from the fisherman. Find : do use right A trig: CAM TOA sino = 7 C = apposite a hypotenuse coso= b = oof tang = a = opp C hyp adj . Sing = 3 C COS O = = 5 C coso. do - 3 dc 134 at C 2 at do = 3 134 (-0.04) at 134 do 34 - 3 ( - 0.04) 5 34 rad /sExample. Water flows into a tank at a rate of 3 cubic meters per minute. The tank is shaped like a cone with a height of 4 meters and a radius of 5 meters at the top. Find the rate at which the water level is rising in the tank when the water height is 2 meters. 5 M dv = 3 ms / min Find : ah at at h= 2 m 4m h 3 V = rah Don't know anything about r ... dv _ IC. 2r dr ht TEz dh INeed more 3 dt 3 dt information use similar A'S : V = TC / shy 2. h 3 = 25 TL 4 to 5 4 V = 251 13 r = 5h 48 3 . 4 = an 25 r dt 4 dv - 25 1 13 /2 dh at dt oh - 12 n 16 25 m Additional Tips and Tricks for Related Rates Problems: . use consistent units! ( Ex: cannot mix cm and m) . Plug in any quantities that are not changing BEFORE taking derivatives. All others plug in after. . Equation you take derivative of should only involve variables whose derivatives you know or are trying to find. 5