Calculus I Homework: Related Rates Page 1

Transcription

1 Calculus I Homework: Relate Rates Page 1 Relate Rates in General Relate rates means relate rates of change, an since rates of changes are erivatives, relate rates really means relate erivatives. The only way to learn how to solve relate rates problems is to practice. The proceure to solve a relate rates problem: 1. Write own the rate which is Given. 2. Write own the rate which is Unknown. 3. Write own your notation an raw a iagram. 4. Fin a formula connecting the the quantities you liste in your Notation. There shoul be no erivatives in this relationship. (a) If necessary, use geometry to eliminate a variable from your formula. 5. Implicitly ifferentiate the formula to get rates of change involve. If you en up with more than one unknown rate of change, you might have to eliminate a variable using geometry (as mentione in the previous step). 6. Solve for the Unknown Rate. 7. Substitute values to etermine the Unknown Rate. 8. Write a concluing sentence. Questions 1. A man starts walking north 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft ue east of P. At what rate are the people moving apart 15 min after the woman starts walking? 2. The altitue of a triangle is increasing at a rate of 1 cm.min while the area of the triangle is increasing at a rate of 2 cm 2 /min. At what rate is the base of the triangle changing when the altitue is 10 cm an the area is 100 cm 2? 3. Water is leaking out of an inverte conical tank at a rate of cm 3 /min at the same time that water is being pumpe into the tank at a constant rate. The tank has height 6 m an iameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, fin the rate at which water is being pumpe into the tank. 4. A baseball iamon is a square with sie 90 ft. A batter hits the ball an runs towar first base with a spee of 24 ft/s. At what rate is his istance from secon base ecreasing when he is halfway to first base? At what rate is his istance from thir base increasing at the same moment? 5. A trough is 10 ft long an its ens have the shape of isosceles triangles that are 3 ft across at the top an have a height of 1 ft. If the trough is being fille with water at the rate of 12 ft 3 /min, how fast is the water rising when the water is 6 inches eep?

2 Calculus I Homework: Relate Rates Page 2 Solutions 1. A man starts walking north 4 ft/s from a point P. Five minutes later a woman starts walking south at 5 ft/s from a point 500 ft ue east of P. At what rate are the people moving apart 15 min after the woman starts walking? Here is a iagram of the situation: The notation I have introuce is: Distance man is from P is x. Distance woman is from Q is y. Distance between them is z. We are given: The man walks with spee 4 ft/s. This means x 4 ft/s 240 ft/min. The woman walks with spee 5 ft/s. This means y 5 ft/s 300 ft/min. The istance between P an Q is 500 ft. The units have been change to ensure they are consistent, in feet an minutes. What is unknown is the rate at which the are moving apart, which is the rate of change of the istance between them, z. To get the relation between x, y, an z we nee to use our iagram. It is easier to see the relation if we reraw our iagram, which I alreay i above. The relation is (x + y) z 2. Implicitly ifferentiate the relation to get a relation between the rates of change. The rates of change are with respect to time t, so we shoul ifferentiate with respect to t. The quantities x, y, an z are all functions of t. [z2 (x + y) ] 2z z ( x 2(x + y) + y ) We solve this for the unknown rate of change: ( z (x + y) x z + y ) To use this equation, we nee to know the quantities x, y, z after the woman has been walking for 15 minutes. Since she starte walking 5 minutes after the man, the man will have been walking for 20 minutes. In 15 minutes, the woman walks y 15 min 300 ft/min 4500 ft. In 20 minutes, the man walks x 20 min 240 ft/min 4800 ft. The istance between them at this time will be z (x + y) ( ) ft.

3 Calculus I Homework: Relate Rates Page 3 The rate of change of the istance between them after the woman has been walking 15 minutes is ( z (x + y) x z + y ) ( ) 100 (4 + 5) 837 ft/min The altitue of a triangle is increasing at a rate of 1 cm/min while the area of the triangle is increasing at a rate of 2 cm 2 /min. At what rate is the base of the triangle changing when the altitue is 10 cm an the area is 100 cm 2? Here is a iagram of the situation: The notation I have introuce is: The altitue of the triangle is h. The base of the triangle is b. The area of the triangle is A. We are given: The altitue is increasing at a rate of h 1 cm/min. The area is increasing at a rate of A 2 cm2 /min. What is unknown is the rate of change of the base, b. The relation between the base an altitue of a triangle is A 1 2 bh. Implicitly ifferentiate with respect to time: A 1 ( h b ) 2 + bh. Solve for the unknown rate of change: b 1 ( 2 A ) h bh. At h 10 cm an A 100 cm 2, b 2A/h 2(100)/10 20 cm. The rate of change of the base at this time is b 1 (2(2) (20)(1)) 1.6 cm/min. 10 The negative sign in our answer means the length of the base is ecreasing. 3. Water is leaking out of an inverte conical tank at a rate of cm 3 /min at the same time that water is being pumpe into the tank at a constant rate. The tank has height 6 m an iameter at the top is 4 m. If the water level is rising at a rate of 20 cm/min when the height of the water is 2 m, fin the rate at which water is being pumpe into the tank. Here is a iagram of the situation:

4 Calculus I Homework: Relate Rates Page 4 The notation I have introuce is: The height of water in the tank is h m. The raius of water in the tank is r m. The rate water is being pumpe into the tank is R m 3 /min. We are given: Water is leaking out of the tank at a rate of cm 3 /min 10 4 (10 2 m) 3 /min 0.01 m 3 /min. The tank has height 6 m an raius at top of 2 m. What is unknown is the rate water is being pumpe in, R. The volume of water in the tank at a specific time is given by V 1 3 πr2 h. We can eliminate one of the variables using similar triangles. r h 2 6 r 1 3 h. The volume of water in the tank is given by V 1 3 πr2 h 1 3 π ( 1 3 h ) 2 h 1 27 πh3. This is the volume for a conical tank of the specific imensions given in this problem. The rate of change of volume of water in the tank is foun by implicitly ifferentiating: V 1 h πh2 9 m/min which must equal R 0.01 m/min

5 Calculus I Homework: Relate Rates Page 5 At h 2 m, h 20 cm/min 0.2 m/min, an we have R h πh2 9 R 1 h πh π(2)2 (0.2) π(2)2 (0.2) m 3 /min The rate water is being pumpe into the tank is m 3 /min when the height of the water is 2m. 4. A baseball iamon is a square with sie 90 ft. A batter hits the ball an runs towar first base with a spee of 24 ft/s. At what rate is his istance from secon base ecreasing when he is halfway to first base? At what rate is his istance from thir base increasing at the same moment? Diagram: x is the istance from home plate. 90 x is the istance from the runner to first base. y is the istance from runner to secon base. z is the istance from runner to thir base. Given the rate of change of istance from home 24 ft/s x Relation from using Pythagorean Theorem: y (90 x) 2.. Nee to fin y.

6 Calculus I Homework: Relate Rates Page 6 Implicitly ifferentiate an solve for y : [y2 ] [902 + (90 x) 2 ] y [y2 ] y x [(90 x)2 ] x 2y y 2(90 x)( 1)x y x) x (90 y (90 x) x (90 x) 2 When the runner is halfway between first an home, x 45 ft an x y 24 ft/s. 5 The answer is negative since the istance to secon base is ecreasing. A similar process for the istance to thir looks like: Relation from using Pythagorean Theorem: z x 2. Implicitly ifferentiate an solve for z : [z2 ] [902 + x 2 ] z [z2 ] z x [x2 ] x 2z z 2xx y x z x x x x 2 When the runner is halfway between first an home, x 45 ft an x z 24 ft/s. 5 The answer is positive since the istance to thir base is increasing. 24 ft/s, so substituting this in we get 24 ft/s, so substituting this in we get 5. A trough is 10 ft long an its ens have the shape of isosceles triangles that are 3 ft across at the top an have a height of 1 ft. If the trough is being fille with water at the rate of 12 ft 3 /min, how fast is the water rising when the water is 6 inches eep? Diagram: We are given the rate of change of the volume is V The unknown rate of change is the water level, h. 12 ft3 /min.

7 Calculus I Homework: Relate Rates Page 7 The volume is V 1 2bh10 5bh. We nee to eliminate the variable b, since we know nothing about it. Use similar triangles to get b 3 h 1 b 3h. No we can substitute this into our volume equation an implicitly ifferentiate: V 5bh V 5(3h)h V 15h 2 V [15h2 ] 15 h [h2 ] h 30h h h 1 30h V When the water level is h 6 inches 1/2 ft eep, the water is rising at the rate of h 1 30(1/2) (12) 4 5 ft/min.