CALCULUS SOLUTIONS FOR WORKSHEET ON PAST RELATED RATES QUESTIONS FROM AP EXAMS

Transcription

1 CALCULUS SOLUTIONS FOR WORKSHEET ON PAST RELATED RATES QUESTIONS FROM AP EXAMS 1. A pape cup, which is in the shape of a ight cicula cone, is 16 cm deep and has a adius of 4 cm. Wate is poued into the cup at a constant ate of cm / sec. (a) At the instant the depth is 5 cm, what is the ate of change of the height? (b) At the instant the adius is cm, what is the ate of change of the adius? 16 h 4 a) The smalle tiangle is simila to the lage tiangle, yielding a popotion 16 h 1 1 h The volume of the liquid in the cone is V h. So V ( h) h h 4 48 Taking the deivative with espect to t, we get dv 1 1 h h We ae looking fo the ate of change of the height,, when h = 5. Futhemoe, since wate is poued into the cup at a constan ate of cm / sec, then we have 5 (5 ) b) Going back to the popotion, we also have h = 4. This gives a volume 1 4 dv d fomula V ( )(4 ). Then 4. Plugging in =, we d d 1 get 4 () 18

2 . A snowball is in the shape of a sphee. Its volume is inceasing at a constant ate of 10 in / min. (a) How fast is the adius inceasing when the volume is 6 in? (b) How fast is the suface aea inceasing when the volume is 6 in? a) The volume is inceasing at a constant ate of 10 in / min implies We want to know how fast the adius is inceasing, which means we want to find dv 10 in / min. The snowball is in the shape of a sphee, so the volume fomula is 4 V dv 4 d dv d 4 In ode to solve fo d, we need to find, because we aleady know that dv 10 in / min. Howeve, can be found with the infomation when the volume is 6 in. 4 V dv d d 10 4 ( ) 6 ( ) Theefoe, 4 10 d d 18 b) The suface aea fomula of a sphee is given by SA 4 dsa d 8 dsa 5 8 ()( ) 18 dsa 0 d.

3 . (1985) The balloon shown is in the shape of a cylinde with hemispheical ends of the same adius as that of the cylinde. The balloon is being inflated at the ate of 61 cm / min. At the instant that the adius of the cylinde is cm., the volume of the balloon is 144 cm, and the adius of the cylinde is inceasing at the ate of cm/min. (a) At this instant, what is the height of the cylinde? (b) At this instant, how fast is the height of the cylinde inceasing? a) The balloon is being inflated at the ate of 61 cm / min means that dv if V is the volume of the balloon, then 61. We need to find the volume of this object, fist. 4 V h Theefoe, if the adius of the cylinde is cm when the volume is 144 cm, then the height can be found by () () h h 108 9h 1 h b) The ate at which the height is inceasing is given by diffeentiating the volume fomula with espect to t. 4 V h dv d d 4 h 61 4 () ( ) ()()(1) ( )

4 4. (1988) The figue shown epesents an obseve at point A watching balloon B as it ises fom point C. The balloon is ising at a constant ate of m/sec, and the obseve is 100 m fom point C. (a) Find the ate of change in x at the instant when y = 50. (b) Find the ate of change in the aea of ight tiangle BCA at the instant when y = 50. (c) Find the ate of change of at the instant when y = 50. dy a) The balloon is ising at a constant ate of m/sec implies that dx. To find, when y = 50, we need to find an equation that elated y, x, and 100. x y 100 x d x dy y 0 d t d t Now we need to find x when y = 50. Using the pythagoean theoem again, we get x x 50 (1 x 50 (5) ) x 50 5 So, then dx ( 50 5) (50)() dx (50)() (50)( 5) 5 b) The aea of the tiangle above is just A 1 (100)( y ) 50 y da dy m 50 (50)() 150 sec c) To find d, we need the equation y tan( ) 100 d 1 dy sec ( ) 100 d cos ( ) 100 adjacent 100 d But cos( ). Theefoe cos ( ) ( ). hypotenuse

5 5. (1994) A cicle is inscibed in a squae, as shown in the figue. The cicumfeence of the cicle is inceasing at a constant ate of 6 in/sec. As the cicle expands, the squae expands to maintain the condition of tangency. (a) Find the ate at which the peimete of the squae is inceasing. (b) At the instant when the aea of the cicle is 5 in, find the ate of incease in the aea enclosed between the cicle and the squae. a) The cicumfeence of the cicle is inceasing at a constant ate of 6 in/sec, implies that dc 6. Since each side is, then the peimete of the squae is given by P = 4() = 8. The cicumfeence is given by C dc d P 8 d theefoe dp d d b) When the aea of the cicle is 5 implies 5. The ate of incease in the aea between the cicle and the squae is given by the fomula A ( ) 4 ( 4) da d ( 4) ( 4)(5)( ) 0 10 ( 4) 0

6 6. (1995) As shown in the figue, wate is daining fom a conical tank with height 1 ft and diamete 8 ft into a cylindical tank that has a base with aea 400 ft. The depth, h, in feet, of the wate in the conical tank is changing at the ate of h 1 ft pe minute. (a) Wite an expession fo the volume of wate in the conical tank as a function of h. (b) At what ate is the volume of wate in the conical tank changing when h =? (c) Let y be the depth, in feet, of the wate in the cylindical tank. At what ate is y changing when h =? 1 a) The volume of the wate is given by V h. But that means I have two vaiables. Howeve, like the fist poblem, 1 h h. So now we can substitute fo, 4 1 h ( ) h V h 7 dv b) We need to find, when h =. dv h ( ) ( h 1) ( 1) c) The volume of wate in the tank is ising at the same ate the wate in the conical tank is daining. Since the volume of the cylindical tank is given by V h The base aea being 400 0, and the adius of the cylindical tank is constant. So the volume of the wate in the cylindical tank is ising at a ate of V (400) h dv

7 7. dz dx dy The sides of the ectangle above incease in such a way that 1 and. At the instant when x = 4 and y =, what is the value of dx? (A) 1 (B) 1 (C) (D) 5 (E) 5 The fist thing we notice is that x y z. This means we can find z and a elation between all the deivatives. dx dy dz x y z and 4 z. So z 5. dy dy ( 4)( ) () (5) (1) dy dy dy dx dy 1 1 So the answe is (B).

8 8. A conical tank is being filled with wate at the ate of 16ft / min. The ate of change of the depth of the wate is 4 times the ate of change of the adius of the wate s suface. At the moment when the depth is 8 ft. and the adius of the suface is ft., the aea of the suface is changing at the ate of 1 ft / min (A) (C) 4 ft / min (E) 16 ft / min (B) 1 ft / min (D) 4 ft / min dv Since the tank is being filled with wate at the ate of 16ft / min, then 16. Since The ate of change of the depth of the wate is 4 times the ate of change of the adius of the d wate s suface, then 4. Since So V h (4 ) dv d 4 d 16 4 () 16 d 1 16 So the aea of the suface is The answe is (C) A da d 1 ()( ) 4