Answers: Calculus 12 Review 3.1-3.4 Student created questions

1. A bee flies along a path which follows the function s(t) = 2t3-45t2+300t-9 at t ≥ 0 where s is in meters and t is in seconds.

a) When is the bee at rest in order to gather pollen?

b) Find the average velocity between t=0 and t=5

c) Describe the motion of the bee.

Answer

a) V(t) = 6t2-90t+300

=3(2t2-30t+100)

=3(2t-10)(t-10)

3(2t-10)(t-10)=0

t=5, 10

b) S(0)= 2(0)3-45(0)2++300(0)-9=-9

S(5)= 2(5)3-45(5)2++300(5)-9

=250-1125+1500-9=616

616-(-9)/5=125 m/s

c) V(t)>0 6t2-90t+300>0, t>10

3(2t-10)(t-10)>0 0

V(t)<0 3(2t-10)(t-10)<0, 5

Motion: the bee flew away from its hive at 010 it flew away again to gather pollen.

2. A girl is riding the roller-coaster. Given the position function .Find the acceleration after 15 seconds.

Solution:

Time(t) = 15s

3. (Physics) Question: A ball is thrown downward from the top of a 30m building. The velocity related to the height can be calculated using the formula h (x)=x2-13x+30. What is the rate of change of the velocity when x=3 and x=6?

Answer: rate of change: [h (6)-h (3)]/(6-3)

= （36-78+30-9+39-30）/3

= -12/3

= -4

-4 m/s

4. The cost function of producing Teddy Bears is

and the revenue is

a) Find the profit function.

b) Find the marginal revenue function and the marginal profit function.

c) Find the profit and the marginal profit when Teddy Bears are sold.

d) What is the cost increased on the -nd Teddy Bear?

Cost increased on the -nd is

5. For a certain rectangle, the length of one side is always three times the length of the other side.

a. If the shorter side is decreasing at a rate of 2 inches/minute, at what rate (with respect to time) is the longer side decreasing?

Use ‘z’ for the longer side, ‘x’ for the shorter side

z=3x

b. At what rate (with respect to time) is the enclosed area decreasing when the shorter side is 6 inches long and is decreasing at a rate of 2 inches/minute?

6. If a baseball is thrown upward with a velocity of 10 m/s from the ground, then the distance, in metres, of the ball above ground level after t second is

a. When does the ball reach its maximum height?

b. How long does it take for the ball to reach the ground?

c. Find the approximate velocity with which the ball strikes the ground

7. The position function of a particle is , where s is measured in metres and t is measured in seconds. Find the acceleration at the instant when the velocity is 0.

8. A fortune teller gives you an equation expressing the future demand function of your doorknob company in terms of doorknobs sold.

p(x)= (5 000 000-x)/5 000

a. What is the revenue function?

R(x)=xp(x)

R(x)= (x)[(5 000 000-x)/5 000]

R(x)= [x(5 000 000-x)]/5 000

R(x)= (5 000 000x-x²)/ 5 000

b. What is the revenue after 500 doorknobs have sold?

(500)[5 000 000-500]/5 000= $499 950

c. What is the marginal revenue after 500 doorknobs have sold?

R(x)= (5 000 000x-x²)/ 5 000

R’(x)=(f’(x)g(x)-g’(x)f(x))/(g(x))²

f’(x) = 5 000 000-2x

g’(x) = 0

R’(x)= [(5 000 000-2x)(5 000)-0]/ (5 000)²

R’(x)= (25 000 000 000-10 000x)/25 000 000

R’(500)= [25 000 000 000- (10 000)(500)]/25 000 000

R’(500)=$999.8 per doorknob

9. The scientist Robbie Ivo formulated the equation expressing the average IQ level at one's particular age (in years)

IQ(t)=-3t+51

a. What is your age’s average IQ level (17)?

IQ(t) = -3t + 51

IQ(17) = -3(17) + 51 = 0

Average IQ is 0.

b. What is the rate of change of your IQ level at age 17?

IQ’(t) = -3

At t=17 it is -3 IQ/year.

10. The following equation represents the displacement as a function of time for a chicken as it flies away from its predator.

C(t) = 4(4+t)² + 4

The following equation represents the displacement vs time of the predator pterodactyl.

P(t) = 2(4+t)² + 3

a. Who is moving faster at t=3 seconds?

C’(t)= 8(4+t)C’(3)= 56km/second

P’(t)= 4(4+t)P’(t)= 28km/second

The chicken is moving faster.

b. What acceleration does the predator have at t=2 seconds?

P’’(t)= 4

At t=2 predator acceleration is 4km/s²

c. At what time will the velocities be equal?

8(4+t)=4(4+t)

32+8t=16+4t

16=-4t

-4 seconds

Never equal

d. Will the predator catch up by t=4 seconds?

C’’(t) = 8km/s²

The predator will never catch up because the chicken's acceleration is bigger.

11. The distance traveled by a bike is given by S=37t2  + 20t

A. When did the velocity reach 30 m/s?

B. At what velocity when the bike has already ride 10s?

Solutions for the problems

a. V(t)= 74 t + 20,t =0.135 s

b. V(t)=74t+20, V(t)=74(10)+20 V= 760 m/s

12. The distance of a moving particle travelled in metres after t seconds is given by a position function s = 2t3- 37t2 + 55.

     a) Find the acceleration as a function of time.

     b) Find the acceleration at t=3.

     c) At which point the acceleration will start having a positive value?

Solutions for the problems

a. s’ = 6t2 - 74t

           s” = 12t -74

     b) s”= -38m/s2

     c) 12t - 74=0

                    t= 37/6 s

13. Dr. Kai tested the concentration, in M, of a drug he invented in a patient’s bloodstream is given by the formula  C(t)=0.12t (t2+2t+2) where t represents the number of hours after the drug is taken.

1. Find the rate of change function        

     Answer: 0.12(t2+2t+2)-1 -(0.24t2+0.24t) (t2 +2t+2)-2

     2)What is the rate of change after 5 hours  

     Answer: 0.12(52+2(5)+2)-1-(0.24(5)2+0.24(5)) ((5)2+2(5)+2)-2= -0.00201607 M/hour

      3)What is the rate of change when the concentration is  9/725 M (0.0124M)

      Answer : 0.0124=0.12t (t2+2t+2)       t=3

          0.12(32+2(3)+2)-1-(0.24(3)2+0.24(3)) ((3)2+2(3)+2)-2= -0.00290657 M/hour

14. A new juice company is planning their factory’s opening. After considering their labour and material costs, they determine the cost of making x cartons of their juice to be     c(x)=0.01x+0.00017x2+1000 .   After a fierce marketing campaign, they found out that their demand is now at p(x)=250000-x120000 in dollars ($).

1. Express their revenue in a function.

2. Express their profit in a function.

3. Find the marginal profit function.

4. How many cartons should the company make to achieve a maximum profit?

Solutions for the problems

1. R(x)=x(250000-x120000)

2. P(x)=x(250000-x120000)-(0.01x+0.00017x2+1000 )

3. P'(x)=250000-2x120000-0.01-0.00034x

4. P'(x)=0 =248800-42.8x   , x=5813.084112 5813 Cartons.

14. Paul throws a ball vertically upward from the surface an unknown planet at a velocity of  24m/s  reaches a height of s = 24t - 0.8t^2  meters in t seconds.a) What is velocity of the ball and the acceleration in terms of time?       velocity : v = s’ = 24-1.6t m/s      acceleration : a = s’’ = -1.6 m/s^2b) Predict how long does it take ball to highest point?      The ball reaches the highest point = velocity equals to 0       v = s’ = 24-1.6t        24-1.6t = 0       t = 15sc) Predict the maximum height that Paul can throw the ball.     The ball reaches the highest point when t = 15s     s = 24t - 0.8t^2     s = 24(15) - 0.8(15)^2     s = 180 md) When does the ball reach half of its maximum height?     half of the maximum is 180m/2 = 90m     s = 24t - 0.8t^2     24t - 0.8t^2 = 90     use quadratic formula     t = 4.4se) What is planet that Paul stands on? Use the acceleration to guess.     The acceleration is a = s’’ = -1.6 m/s^2     Using common sense in Physics, the moon’ s gravitational field is 1.6 m/s^2.     Therefore, Paul is standing on the Moon

15. A position function is given by d=4cost+t2, where d represents displacement in meter and t represents time in second. Shown that, during the movement of the car, there is at least one point where the instantaneous velocity is zero.

d ’=-4sint+2t

0=2sint-t

f(x)=2sinx-x

f(π/2)=2- π/2>0

f(π)=0- π/2<0

f(x)=0 for some x in (π/2, π)

16. The cost of manufacturing automobiles is given by the equation C(x)=75 +15x -0.05x2 + 0.004x3. Find the marginal cost of producing 20 automobiles.

C’ (x)=15 -0.1x + 0.012x2

C’ (20)=15 -2 + 4.8 =18

17. A cylinder clay is horizontally stretched so that its radius increases while the height stays constant. Find the rate of change of the volume with respect to the radius when the radius is 5cm and height is 15cm.

V=πr2h

V= 15πr2

V’= 30πr=150π

18. The motion of a rocket is described by the equation:

S = - 69t2 + 21t + 420

Where S is in upward meters and t is in seconds.

a) What is the rocket’s velocity at 0.5 hours?

b) When does the rocket’s velocity reach -2km/h?

c) What is the velocity when the rocket lands?

Solution:

a) V(t) = S’ = (- 69)(2t) + 21

V(t) = -138t + 21

V(t) = - 248379 m/s

b) V = - 2 km/h = = -

-= -138t + 21

-138t = -- 21

T =

c) - 69t2 + 21t + 420 = 0

T =

T = 2.62 or - 2.32

V(t) = -138t + 21

V(2.62) = - 341.117 m/s

18 b. Find acceleration when x = 5 and S’ equals the velocity in meters (s) per seconds (t)

S’ = (7 - x)2 (x2)8

Solution:

S” = (7 - x)2 (16x15) + 2(7 - x)(-1)(x)16

= 16x15(7 - x)2 - 2x16(7 - x)

= 16(5)15 (7 - 5)2 - 2(5)16(7 - 5)

= 1.34 x 1012 m/s2

19. The population of a bacterial colony after t minutes is given by the equation

P = 2218 + 200t + 20t2 + 2t3

a) Find the average rate of population growth between t = 6 and t = 12

b) Find the instantaneous rate of population growth at t = 250

Solution:

a) m =

m = =

m = 6384 bacteria/minute

b) P = 2218 + 200t + 20t2+2t3

P’ = 200 + 40t + 6t2

P’ = 200 + 40(250) + 6(250)2

P’ = 385200 bacteria/minute

20. A company pays 800 dollars in production cost before it even begins to manufacture any of its items. Then for each item produced, the production cost increases by 10 dollars. The company estimates the demand function for its products from a group of 100 people wanting the product but for each product produced, the demand goes down by one person.

a) What is the production cost function for x items in dollars?

b) What is the estimated demand function for x items?

c) Find the marginal profit at a production level of 30 items.

Solution:

a) C(x) = 800 + 10x

b) p(x) = 100 - x

c) R(x) = xp(x)

P(x) = R(x) - C(x)

P’ = marginal profit

R(x) = 100x - x2

P(x) = 100x - x2 - 800 + 10x

P(x) = -x2 + 100x - 800

P’(x) = -2x + 100

21. An IS-7, a Soviet heavy tank which is famous for its significantly strong armor, rolls out to find an enemy tank. When people in the tank found an enemy, they drove the IS-7 to a place where he can lower its gun. The enemy tank fires his shells to IS-7 and those crews inside IS-7 is thinking how they can stop damage from happening when the enemy fires at them. The amount of damage caused by the enemy tank in percentage, can be reduced when the gun depression (the degrees in which a tank’s gun can lower to.) gets lower. The function of damage (unit: HP) proportional to the gun depression angle (°) is the function below.

d(θ) = -1.5θ^2 -11θ +100 (θ≥0)

a) Find the average rate of change of damage in HP when gun depression in degrees changes from 1° to 3°.

b) Determine how fast the damage changes with respect to gun depression when θ=4°.

Answer:

a) (d(3) – d(1)) / (3-1) = -17 HP/degree

b) d’(θ) = -3θ -11 d’(4) = -23 HP/degree

c) A cyclist is riding a bicycle uphill. The distance(in m) that travelled by a cyclist is given by d=12t²+4,find the velocity of the cyclist when t=3 min.

d=12t²+4

d’=12×2t+0=24t

t=3,

v=24×3=72 m/min

22. The height of a cylindrically shaped rubber tube, with a volume of 81π cm₃, is stretched at a constant rate of 8 cm/min. The volume of the rubber tube does not change as it stretches vertically. Find the rate of change in radius when the height is 15cm.

V=πr2h

r2=v/πh=81/9 r=3

dv/dt=2πrh x dr/dt + πr2dh/dt

dr/dt=-6

23. The population of a bacteria colony after t hours is given by the function .a. Find the average rate of growth between 5 and 8 hours.Solution: =7225000 =95081008 bacteria/hourb. Find the instantaneous rate of growth at t=2 hourssolution: At t=2, the growth rate is 191296 bacteria/hour. 24. A group of scientists are researching how they can harness quantum energy from a particle. The scientists managed to figure out a position with respect to time function for the particle; where t is the time in seconds and is the position or displacement. The scientists realize that in order to determine the correct spin of the particle, they would have to capture the motion of the particle with a high-speed camera when the particle has a positive acceleration. A. At what time should the scientists try to capture the particle?solution: (this is the formula for acceleration)Then solve for t and you get t=7.270771696 and 0.229when or , the acceleration is positive.Therefore, the scientists should try to capture the motion of the particle between 0 and some time less than 0.229 s, and after 7.270771696s has elapsed.B. Scientists later found that de wey will be shown if the particle moves at constant velocity, at what time should the scientists try to capture the motion of the particle if they want to see de wey?solution: When the acceleration is 0, the velocity will be constant. Therefore, the answer is t=0.229s and t=7.270771696s. But it is only at these two instances.

b. The average satisfaction from sleep a student gets per night instandard unit kMS(kilomorning*sleep) can be graphed in relation to thestudent's grade in percentage. This function with zero scientificreasoning is

a) Find the instantaneous rate of change in amount of sleep at a grade of 80%

b) For a student in Prince of Wales, calculate the hours of sleep astudent with a grade of 145% gets.

Answera) when g = 80; -0.03913118961b)DNEGrade cannot go over 100%

b. If superman throws a volleyball straight upwards, its initial velocity is 92 m/s and its height (meters) in terms of t (seconds) is

h(t)=92t-4.9t²

a. When does the velocity reach 80 m/s?

b. Find the velocity after 1, 3, 5 seconds

c. When does the ball reach maximum height?

d. What is the maximum height?

e. When does it hit the ground?

Solutions

a. V= 92 - 9.8t

80= 92 - 9.8t

t= 1.2 seconds

     b) V= 92 - 9.8t

           V= 92 - 9.8(1)= 82.2 m/s

V= 92 - 9.8(3)= 62.6 m/s

V= 92 - 9.8(5)= 43 m/s

     c) V= 92 - 9.8t

          0= 92 - 9.8t

          t= 9.4 seconds

     

     d) from question c use the time at max. Height

          h(t)=92t-4.9t(squared)

          h= 431.8m

e) h(t)=92t-4.9t(squared)

       0=92t-4.9t(squared)

       t= 0, t= 18.8seconds

b. The position of a spaceship is given by s = 7t3 - 4t2 + 32t - 9, where s is measured in metres and t is measured in seconds.

a. Find the acceleration at 4 s.

a. When is the acceleration 0?

a. What is the velocity of the spaceship when the acceleration is 244 m/s2?

Solution:

a. s(t) =  7t3 - 4t2 + 32t - 9

v(t) = 21t2 - 8t + 32

a(t) = 42t - 8

a(4) = 42(4) - 8

a(4) = 160 m/s2

b. a(t) = 42t - 8

0 = 42t - 8

42t = 8

t = 0.19 s

c. a(t) = 42t - 8

244 = 42t - 8

42t = 252

t = 6 s

v(t) = 21t2 - 8t + 32

v(6) = 21(6)2 - 8(6) + 32

v(6) = 740 m/s

28. The following data was obtained for a reaction where zinc metal (Zn) was reacted with HCl.

Time (min)

Mass of Zn (grams)

0

31.0

1

24.6

2

20.2

3

17.4

Calculate the average rate of the reaction (gram/second) within the first minute. Round answer to three decimal places.

Answer:

△y△x﹦ y₂ - y₁x₂ - x₁﹦ f(x₂) - f(x₁)x₂-x₁ plug in (0, 31.0) and (1, 24.6).

24.6 - 31.01-0﹦﹣6.41﹦- 6.4 g/m

﹣6.4g1min×1min60secs﹦ 0.107 g/s

The average rate of reaction within the first minute is 0.107 g/s.

29.

A hot dog company determines that the cost of manufacturing golden frisbees is:

C(x) = 42000 + 15x + 0.14x²

a. Find the marginal cost function.

b. Find the marginal cost at a production level of 400 items.

c. Find the cost of producing the 401st item.

Answers:

a. 15 + 0.28x

b. 15 + (0.28)(400) = $127/item

c. [42000 + 15(401) + 0.14(401²)] - [42000 + 15(400) + 0.14(400²)] = $127.14

30. A dragon boat crosses the finish line and everyone stops paddling. The dragon boat slowly comes to a stop. The position of the dragon boat is s(t) = -0.4t2 + 5t where s (in meters) is a function of t (in seconds).

a. How fast was the dragon boat moving right after everyone stopped paddling?

b. When does the dragon boat stop moving?

c. Find the distance the dragon boat drifted from the finish line until the dragon boat stops moving.

SOLUTIONS

a. s’(t) = -0.8t + 5

s’(0) = -0.8(0) + 5

s’(0) = 5 m/s

     b) s’(t) = -0.8t + 5

           0 m/s = -0.8t + 5

           -5 = -0.8t

           t = 6.25 seconds

     c) s(t) = -0.4t2 + 5t

           s(t) = -0.4(6.25)2 + 5(6.25)

           s(t) = 15.625m

32. A rocket headed for the moon takes off from Earth. As it leaves the atmosphere, it accelerates until the engines are turned off. Its position while accelerating can be described by the function s(t) = 2t4 + 3t3 when s is measured in metres, t is measured in seconds, and t ≥ 0.

a. Find the rocket’s velocity as a function of t.

b. Find the rocket’s acceleration as a function of t.

c. At time t = 30:

i. What is its velocity?

ii. What is its acceleration?

d. Find t when the acceleration is 10,000 m/s2. Round to the nearest one hundredth of a second if necessary.

SOLUTIONS

a. s(t) = 2t4 + 3t3

           s’(t) = 8t3 + 9t2

b. s’(t) = 8t3 + 9t2

           s”(t) = 24t2 + 18t

c.

i) s’(30) = 8(30)3 + 9(30)2

  s’(30) = 224,100 m/s

           ii) s”(30) = 24(30)2 + 18(30)

   s”(30) = 22140 m/s2

     d) s”(t) = 24t2 + 18t = 10000

           0 = 24t2 + 18t - 10000

           t = -20.79, 20.04

           t ≥ 0, t = 20.04

           t = 20.04 s

   

33. With permission and certification Alexander Tsang reintroduced a pack of 10 lynxes back into Garibaldi National Park to control the deer population. Over the next 100 years the government of Canada monitored the pack and documented its population numbers. Today, you, the scientist will answer questions given in order to better help understand the reintroduction process of wolves.

1. What is the average rate of growth per year between years 0 and year 20?

a. For deers?

a. For lynxes?

0. What year saw the greatest drop in population for deers?

a. What population was it 5 years before the great drop.

b. Find the average growth per year in those 5 years

0. Was Alexander Tsang successful in controlling the deer population?

SOLUTIONS

1. (y2-y1)/(x2-x1)

a. (175-202)/(20-0)

i. -27/20

i. -1.35

1. (16-10)/(20-0)

ii. 6/20

ii. .3

0. Year 40

a. 165

b. (y2-y1)/(x2-x1)

i. (130-165)/(40-35)

ii. -35/5

iii. -7

0. Yes Alexander Tsang was successful

(x) years

(y) lynx population

(y) deer population

0

10

202

5

10

202

10

12

190

15

14

180

20

16

175

25

18

172

30

19

169

35

18

165

40

18

130

45

11

122

50

10

120

55

10

121

60

12

124

65

11

125

70

10

122

75

11

120

80

11

124

85

12

119

90

12

113

95

11

110

100

11

112

34. A river flows through a canyon formed of sedimentary rock. The amount of rock eroded is represented by the function d(t) = -0.0004t2 + 0.08t  where d is the vertical distance of rock eroded in meters, and t is the amount of time passed in years.

This function has a restricted domain:  0 ≥ t ≥ 100

a. Find the instantaneous rate of erosion in terms of t.

b. Find the average change of rate of erosion between years 5 and 12.

c. What vertical distance of rock is eroded by the river in total?

SOLUTIONS

a. Vertical distance of rock eroded is d(t) = -0.0004t2 + 0.08tTherefore, rate of erosion is d’(t) = -0.0008t + 0.08

b. Vertical distance of rock eroded is d(t) = -0.0004t2 + 0.08t

Rate of erosion is d’(t) = -0.0008t + 0.08

( y2 - y1 ) / ( x2 - x1 )

(0.0704-0.076)/(12-5) = -0.0008 m/y

ALTERNATE SOLUTION:

d’’(t) = -.0008 m/y

     c) Vertical distance of rock eroded is d(t) = -0.0004t2 + 0.08td(100) = -0.0004(100)2 + 0.08(100)

d(100) = 4 m

35. The cost function for the production of frappuccinos at Starbucks is:

C(x) =7000+2x+0.01x2

where x is the number of frappuccinos made.

A. Find the marginal cost function and determine the cost when there are 300 frappuccinos made.

B. How many frappuccinos are being produced if the marginal cost is $6.50

SOLUTIONS

A. C’(x) = 2+0.02x, $8

B. 225 frappuccinos

36. Bush produces and sells aircrafts and fixed cost of the company is $30000 and the variable cost is $500 per unit and sells the aircraft for $1000 per unit.

a) Find the total cost function

b) Find the revenue function

c) Find the profit function and determine when 1000 units are sold

d) Have many units have to be produced to yield a profit of $60000

Answers:

a)30000 + 500x

b) 1000x

c) R(x) – C(x) = 1000x – (30000+500x)= 1000x-30000-500x=500x-30000

x=10000 500(10000)-30000 = $4970000

D) p=$60000 60000=500X-30000 X= 180 units

37. A ball thrown vertically upward from the surface of the moon at a velocity of 20m/s reaches a height of S(t)=20t-2t2 .

a) Find the Find the ball’s velocity and acceleration as functions of time.

S’(t)=V(t)=20-4t V’(t)=A(t)=-4 b) What is the velocity of the ball after 3 second V(3)=20-4*3=8m/s c) What is the ball’s acceleration when t = 2 second A(2)=-4 The ball is traveling at constant acceleration d) What is the ball’s velocity when it reaches the ground? S(t)=0=20t-2t2 t(20-2t)=0 t=0(starting point) or t=10s V(10)=20-4(10)= -20m/s e) When does the ball reach the maximum height? When v=0 the ball reach the maximum height, thus, V(t)=0=20-4t t=5s

38. If Eric is thrown upward with an initial velocity of 10m/s, then its height after t seconds, in meters, is      h=10t-4.9t^2

a) Find the velocity after 1s, 2s, and 3s,

b) When does Eric reach his maximum height?

c) Can he reach 6m?

d) With what velocity does he hit the ground?

e) Will he break his legs?

a) 0.2, -9.6, -19.4 b) 1.02 sec c) No d) -10.001 e) No, he is strong

39. A student in Ms. Lau’s math class talks too much. One day, Ms. Lau is irritated and throws a chalk to her annoying student at a high speed of 32m/s. Find the acceleration of the chalk with given equation, s=32t-4.9t^2 in respect to t.

s’=32-9.8t

40. The number of germs in Ms. Lau's class is defined by the equation 15t^2+70t+1000.

a) Find the instantaneous rate at which the number of germs increase at t=10 and t=15?

b) Find the average rate at which the number of germs increase during the 5 minutes from t=10 to t=15?

a) 370, 520 b) 445

41. Ms. Lau’s T-Shirt shop has a production cost of 666+6x+0.6x^2, x is the number of shirt she produced.

a) What is the marginal cost function? and how much is it if x=1?

b) How many shirts did she produce when the marginal cost is $12?

c) If she sells it at a price of $18, when should she stop producing?

d) Will you buy the t-shirt?

a) 6+1.2x, 7.2 b) 5 c) 10 d) yes

42. Eric runs after Juliett and his position as a function of time is expressed by

s=2t+5t2-23+3t+6-9t2-t.

a) Find his velocity in terms of time t

(-10t^2-30t-4)/(t^2-2)^4 -81t^2-114t-6

b) Find the velocity of the virus at t=5 seconds

-404/(23)^4 -2601

c) Notice she gets sick from Eric, give a possible explanation why.

That’s pretty cool.

43. The population of bacteria on Cayden’s marvellously soft hand travels very quickly and its position given by a function of time is given by the formula

s= 13t35t2-t

a) What is the instantaneous acceleration?

1300t^3+1/(4t√t)

44. Cayden’s spherical head is being inflated. Find the rate of change of the volume with respect to the radius when the radius of the head is 100cm.

V(r)=43r3

40000cm3/cm

45. Ms. Lee produces x batches of children and has determined that the demand from her husband is p(x)=(48-x)/12x.

Determine the batches of babies Ms. Lee must produce in order to maximize its profit.

p’(x)=-1/12

When p’(x)=0

Ms. Lee can have infinite amounts of babies.

*Is Cayden really Ms. Lee’s son?

46. A projectile is shot vertically above from a few metres above the ground. The function of the projectile’s vertical displacement from the ground, , is provided below:

Assuming that up is positive, answer the following questions:

a) Find the average velocity of the object between and . (3 marks)

b) Determine the equation for instantaneous velocity. (3 marks)

c) Find the instantaneous velocity when . (2 marks)

d) Find the average acceleration between and . (2 marks)

e) Find the equation for instantaneous acceleration. (3 marks)

f) Find the instantaneous acceleration when . (2 marks)

g) At what time does the velocity become 0 in midair? (5 marks)

h) Find the velocity of the projectile when it lands. (5 marks)

/25

47. The population of bacteria colony ( bacteria) after hours is given by the function:

a) Find the average growth rate between and . (2 marks)

b) Find the instantaneous growth rate at (4 marks)

/6

48. A water bottling company has determined its demand function is:

Its cost function is:

Determine how many bottles of water it needs to bottle to maximize profit. (3 marks)

/3

-

SOLUTIONS:

46

a)

*Significant figures are not necessary.

b)

c) Using the result from b),

d) Using the result form b),

e) Using the result from b),

f) Using the result from e),

g) Since ,From b) we know that:

h) When the projectile lands,

Reject -0.09sUsing the result from b)

47

a)

b)

48 Since

And

Reject

49. The distance travelled by a moving object is S(x)= where x is in hours and S in metres. Find its velocity after 1.1 hour. (round to nearest whole number)

50. The position of a moving object is given by the function S(t)=, where S is in metres and t in seconds. Find the object’s average acceleration between t=0 & t=5, and its instantaneous acceleration at t=4.

51. Two people are blowing perfectly spherical balloons. Person A is blowing at a rate such the radius of the sphere increases by 0.7cm/s

a) What is the general formula of the rate of change in Volume of Person A’s balloon?

b) Radius of person B’s balloon is given by r(t)=. What is the instantaneous rate of change in volume when t=3?

c) Assume person A 5start blowing at r=0. At what time are their balloons the same size? At that time, who’s balloon is growing quicker?

52. A company sells toys whose revenue is given by R(x)=, where R is in thousand dollars and x is in thousand units.

a) what is the maximum amount of revenue they can obtain?

b) If the cost of each thousand of toy is given by C(x)=0.2, what is the max output they should produce at to maximize profit? Solve in two ways.

Answer Key:

49. V(x)=S’(x)=7. Substitute in x=1.1, V(1.1)=S’(1.1)=524km/h.

50. Average acceleration====13

Instantaneous Acceleration=S”(t)=. Substitute in 4 to get a=22

51. a) Rate of change in volume==

b) Rate of change in volume== Substitute in t=3 to get =284752π.

c) If both start at 0 then we know radius of person A’s balloon is given by R(t)= (using antiderivative and C=0). Therefore the Volume of each person is both given by . Place one expression on either side of an equal sign and substitute in r= & r=, and solve for t. Final answer t=2.019s. substitute t=2.019 into the of person B, and get the value of 22.458 which is greater than 0.7. So at the time when both persons’ balloon are the same volume, person B’s balloon is growing quicker.

52. a) At maximum slope=0 so R’(x)==0, solving to get x=0.232 & x=1.434. Use values close to either left and right of those 2 points of X and we can see that in the derivative function, values left of 0.232 is negative, between 0.232 and 1.434 is positive, and right of 1.434 is negative. At a maximum, the slope should go from positive-to-zero-to-negative. Therefore x=0.232 is a minimum and x=1.434 is the maximum. Then substitute in x=1.434 to get R=6.065.

b)

Method 1: Profit=Revenue-cost, therefore the Function of profit is given by 0.2 . Make that equal to 0 and solve for x. We get x=0.236 & x=1.414. Using the same identification method in part A we know the maximum is x=1.414

Method 2: Profit is maximized when marginal cost is equal to marginal revenue. Marginal cost and revenue are represented by C’(x) & R’(x) respectively. Make and solve for x. We get x=1.414

53. The distance (measured in metres) that it takes Joe to hoverboard to school after t minutes can be modelled by the equation s(t) = t³ + 4t² - 9t + 26

a. What is the equation for velocity?

b. Find the velocity at t = 5 min

c. Find the acceleration at t = 15 min

54. The population of exceeding students at Prince of Wales Secondary after t terms can be modelled by the equation e(t) = 204 - 30t + 27t²

a. Find the instantaneous growth rate after t terms

b. Find the instantaneous growth rate after 2 terms

55. The Prince of Wales Cafeteria produces x batches of hot dogs and has determined its demand as the function p = 3 + 0.5x and the cost function as C(x) = 10 - 3x + 1.5x² + 2x³

a. Determine the cost to make 4 batches of hot dogs

b. Determine the revenue function

c. Determine the number of batches the cafeteria must produce to maximize its profit

Solutions:

1.

a) s’(t) = 3t² + 8t - 9

b) s’(5) = 3(5)² + 8(5) - 9

= 106 m/min

c) s”(t) = 6t + 8

s”(15) = 6(15) + 8

    = 98 m/min²

0.

a) e’(t) = -30t + 54t

b) e’(2) = -30 + 54(2)

       = 78 students/term

0.

a) C(4) = 10 - 3(4) + 1.5(4)² - 2(4)³

      = $150

    b) R(x) = xp(x)

        = x(3 + 0.5x)

        = 3x + 0.5x²

    c) P’(x) = R’(x) - C’(x)

        0 = R’(x) = C’(x)

        = 3 - x = -3 + 3x + 6x²

        = 6x² + 2x - 6

    -2 ± 2² - 4 (6) (-6)2 (6)

    = -2 ± 14812

    = ~ 0.85

56. A 30 cm long cylindrical rod with a radius of 1 cm is stretched at a rate of 2 cm/s. Its volume stays constant during the process. Assuming the rod is stretched out evenly, its radius r after t seconds can be expressed with this equation (V = volume):

Find the instantaneous rate of change of radius r after 8 seconds.

= -0.018 cm/s

57. Given , t≧0 be a position for an object moving on -axis.

a) Find the velocity function.

b) Find the initial velocity.

c)When is the object at rest?

d) What is the total displacement for

, right 5

58. A particle moves along a line so that its position at any time t≧0 is given by the function where s is measured in meters and t is measured in seconds.

a) Find instantaneous the velocity.

b) Find the acceleration of the particle.

c) When is the particle at rest?

t or 2

d) At what values of t does the particle change directions in the motion of the particle?

A: and 2

59.The amount of bacteria is given as , t in minute .

a) Find the initial amount.

b) Find the function of bacteria change.

c) Find the initial rate of change.

bacteria/minute

d) Find the rate of change when .

bacteria/minute

60. Given revenue . Cost function .

a) Find the profit function.

b) Find the marginal revenue function and marginal cost function.

① Marginal Revenue Function

② Marginal Cost Function

d) What is the value of , if the marginal revenue function equals the marginal cost function?

61.   A steak is climbing a mountain, its distance in meters after t second is d = x2 + 2

a)      What is the average rate between 1 second and 2 seconds?

b)     What is the instannous velocity?

c)      What is the velocity after 5 second?

d)     What is the acceleration and what does the answer mean?

a)      X = 1 Y = X2 + 1 = 1 + 2 = 3

X = 2 Y = X2 + 1 = 4 + 2 = 6

b)     d = X2 + 2

V = 2x

c)      V= 2x = 2 x 5 = 10 m/s

d)     V = 2x

A= 2

  Acceleration is constant at 2 m/s2

62.   A car is falling from a building, the acceleration limit is 16m/s2, the velocity of the car is determined by v = X3 + 2x +1. Can cops fined the car after 6 seconds for going over the acceleration limit?

a)       V’ = 3x2 + 2;  x = 6 ; V’ = 110m/s2   The car is over the acceleration limit, cops can stop and fine the car.

63. The displacement of Bob’s brain cells in (m) during Calculus test due to time (t) in minutes is given by the function:

d(t) = -t^2+4t+4

a)       Calculate the instantaneous velocity of Bob’s brain cells at 2 minutes.

b)       How long does it take for his brain cells to reach the instantaneous velocity of 2 m/s?

c)       How long does it take for his brain cells to come to a stop?

Answer:

a)       V(t) = -2t+4   Sub in t=2

-2(2)+4 = 0 m/s

              b) 2 = -2t + 4

                             -2t = -2

                              t = 1 min

c. 0 = -2t +4

t = 2 mins

64. If a ball is thrown vertically upward with a velocity of 80 m/s, then its height after t seconds is s = 160t − 32t^2 .

a. What is the maximum height reached by the ball?

b. What is the velocity of the ball when it is 192m above the ground on its way up?

c. On its way down?

Answer:

a. v(t) = 160 - 64t

when v= 0, t = 2.5 sec.

Maximum height: s(2.5) = 200m

     b) s(t) = 192 = 160t - 32t^2

t = 2 sec (way up) and 3 sec (way down)

v(2) = 160 - 128 = 32 m/s

     c) v(3) = 160 - 192 = -32 m/s

65.The gravitational force between one object with a mass of 50 kg and another object with a mass of 10kg is represented by

F=

Where force F is in Newtons, and r is the distance between the two masses in meters. Determine what the instantaneous rate of change of F must be with respect to r (in Newtons/meter) when the distance between the objects is 0.5 m and is increasing

5.338x

66. When HX Company sells radio at $50 a piece, they produce and sell 3000 of them per month. For every $1 increase in price, the number of radio they sell decreases by 15. Assume that the fixed production costs are $50,000 and the variable costs are $30 per radio.

(a) Find the linear demand function q = D(p), where p is a price of a unit and q is the number of radio made and sold.

q − 3000 = −15(p − 50) ⇒ D(p) = −15p + 3750

(b) Find the cost function C(q) as a function of q, and then express it as a function of p.

C(q) = 30q + 50 000 C(p) = −450p + 162 500

(c) Find the revenue function R(q) as a function of q, and then express it as a function of p.

R(q) = pq = − 1 q2 + 250q 15

R(p) = pq = −15p2 + 3750p

67. An observer stands 700 ft away from a launch pad to observe a rocket launch. The rocket blasts off and maintains a velocity of 900 ft/sec. Assume the scenario can be modeled as a right triangle. How fast is the observer to rocket distance changing when the rocket is 2400 ft from the ground?

864ft/sec

68. The position s of a particle at time t is given by s(t) = 3t² - 2t

1. What is the particle's velocity at t = 0?

3t² - 2t

6t - 2

6(0) - 2

= -2 m/s

b) What is the particle's velocity at t = 5?

3t² - 2t

6t - 2

6(5) - 2

= 28 m/s

c) What time does s = 0?

3t² - 2t

6t - 2

6t - 2 = 0

6t = 2

⅓ = t

d) What is the particle’s acceleration? What is one observation you can make of it?

3t² - 2t

6t - 2

s’’(t) = 6

The acceleration is constant

69. The population of a bacteria colony, called the doshes, after t hours is given by n = 10000 + 60t + 15t² + 5t³. FInd the growth rate of doshes after 4 hours.

Rate of growth: n’ = 60 + 30(t) + 15t²

= 60 + 30(4) + 15(4)²

= 60 + 120 + 240

= 420 doshes after 4 hours

70. A company determines that the cost, in dollars, of producing x items is

c(x) = 20 000 + 24x + 0.048x²

1. Find the marginal cost function

c’(x) = 24 + 0.096x

b) Find the marginal cost at a production level of 50 items

c’(50) = 24 + 0.096(50)

= 24 + 4.8

= 28.8

c) Find the cost of producing the 51st item

c(50) = 20 000 + 24(50) + 0.048(50)²

= 21 320

c(51) = 20 000 + 24(51) + 0.048(51)²

= 21 348.848

c(51) - c(50) = 21 348.848 - 21 320

= 28.848

The 51st item will cost $28.85

71.

   

Justin lives right across a 1.7km long river from Ellen. One Sunday morning, Ellen calls Justin over to her place. Justin hops on a boat and starts paddling. His displacement as a function of time is given by d(t)=t3-24t2-2t+1

A) What is Justin’s instantaneous velocity. Neglect current..

Instantaneous velocity = 1st derivative of the position time graph

    d'(t)=3t2-48t-2

B) What is Justin’s displacement after t= 30 seconds

    d(t)=(30)3-24(30)2-2(30)+1=5241

5241 meters           

C) What is his acceleration at t=60 seconds

Acceleration= 2nd derivative of position time graph

d''(t)=6t-48

d''(t)=6(60)-48=312

72.  Vivien’s face (a spherical ball) inflates at a rate of 24π cm3/sec. Find the rate of change in its surface area when the radius is 5cm.  

V=43r3

A=4r2

Solution: dVdt =24 π  r=5

dVdt =4π r2 drdt

24=4π r2 drdt

drdt=625

dAdt=8π rdrdt

dAdt=40π 5cm^2/sec

73. Jongho is 2 seconds away from being deported back to North Korea. He tries to run away from his people, but struggles. This is described as the function J(t)=[6t3 + 3t2 - 9t +5] in terms of t. Find the instantaneous velocity sad Jongho is running at.

J(t) = 6t3+3t2-9t+5

J’(t)= 18t2+6t-9

18(2)2+6(2)-9

=75m/s.         Jongho is running at 75m/s

74. Elisabeth(Elsie B) running away from her Singaporean father who has a coat hanger in his hands ready to strike her for failing calculus. The distance from her father in meters can be described in the 20t3 - 7t2 + 9t- 23, in terms of t (seconds)

A. What is the instantaneous velocity and acceleration when t=15 seconds

20t3 - 7t2 + 9t- 23  v’(t) = 60t^2-14t+9v’(15)=60(15)^2-14(15)+9

B. When she gets caught and she gets beat, how fast was she running?

0=20t3 - 7t2 + 9t- 23  x1= 1.0178x2= -0.33393x3=-0.33393

75.

76. The relative displacement of a particle at time t is defined by the function

f(t) = -2

What is the instantaneous acceleration of the particle at t=2?

- Answer

Acceleration = 2nd derivative of the function

Therfore

f’(t) = -6+12t

f’’(t) = -12t + 12

f’’(2) = -12(2) + 12 = -12

-12m/

78. Below is a sketch of a Ferris wheel,a device carrying passengers around the rim of the wheel.

(a) The circular Ferris wheel has a radius of 20 meters and is revolving at a rate of 6 radians per minute. Determine how fast a passenger on the wheel is going vertically upward when the passenger is at point A, 12 meters higher than the center of the wheel and is rising.

(b) The operator of the Ferris wheel stands directly below the center such that the bottom of the Ferris wheel is level with his eye line. As he watches the passenger his line of sight makes an angle alpha with the horizontal. Find the rate of change of Alpha at point A