Accumulation

AccumulationAccumulation

If “V” represents the total volume of a solid

(in cm3) and t represents time (in minutes),

then what does represent? What are

the units?

dV

dt

…and what does represent?b

a

dV

dt

( ) ( )b

a

dVV b V a

dt

or total change in volume

Volume at time bVolume at time a

Total change is also referred to as displacement.

Example: The rate at which water in Example: The rate at which water in

a pond is evaporating is given bya pond is evaporating is given by

How much water has evaporated How much water has evaporated during the first 10 days? during the first 10 days? Include units in your answer.Include units in your answer.

( ) 1 /tW t e liters day

litersW( t )dt .10

0

10 452

Another example…Oil is leaking from a storage tank at the rate 0.3( ) 4000 tR t e

(a) How much oil as leaked in the first 5 days?

(b) How much oil has leaked on the time interval [5, 10]?

Liters/day

2000 Question 4 (NO Calculator allowed)Water is pumped into an underground tank at a constant rate of 8 gallons per minute. Water leaks out of the tank at the rate of (t + 1)1/2

gallons per minute, for 0 < t < 120 minutes. At time t = 0, the tank contains 30 gallons of water.

• What are the key words in the stem?

• What’s different about the rates other than one’s constant and the other is not?

• What about the 30 gallons at t = 0? What does that hint at?

The parts:What are they asking you to do?

(a) How many gallons of water leak out of the tank from time t = 0 to t = 3?

Part (b) is pretty easy, but it’s really helping you to think about the next part.

(b) How many gallons of water are in the tank at time t = 3 minutes?

Part (c) can be done by continuing from part (a). What kind of function do you end up with?

(c) Write an expression for A(t), the total number of gallons of water in the tank at time t.

(d) At what time t, for 0 < t < 120, is the amount of water in the tank a maximum? Justify your answer.

• Part (d) is a max/min problem based on the answer to (c).

o How do you write a justification for the maximum?

What needs to be said or shown?

o How does the FTC fit into this?

2002 Question 2 (Calculator Active) The rate at which people enter an amusement park on a given day is modeled by the function E defined by

2

15600( )

24 160E t

t t

The rate at which people leave the same amusement park on the same day is modeled by the function L defined by

2

9890( )

38 370L t

t t

Both E(t) and L(t) are measured in people per hour and time t is measured in hours after midnight. These functions are valid for 9 < t < 23, the hours during which the park is open. At time t = 9, there are no people in the park.

Compare and contrast the given information in this question with 2000 #4

(a) How many people entered the park by 5:00pm (t = 17)? Round your answer to the nearest whole number.

(b) The price of admission to the park is $15 until 5:00pm. After 5:00pm, the price of admission to the park is $11. How many dollars are collected from admissions to the park on the given day? Round your answer to the nearest whole number.

(c) Let The value of H(17) to the nearest whole number is 3725. Find the value of H’(17) and explain the meaning of H(17) and H’(17) in the context of the park.

9

( ) ( ( ) ( )) 9 23t

H t E x L x dx for t

(d) At what time t, for 9 < t < 23, does the model predict that the number of people in the park is a maximum?

Concert tickets went on sale at noon (t = 0) and were sold out within 9 hours. The number of people waiting in line to purchase tickets at time t is modeled by a twice differentiable function L for 0 < t < 9. Values of L(t) at various times t are shown in the table.

(a) Use the data in the table to estimate the rate at which the number of people waiting in line was changing at 5:30 pm (t = 5.5). Show the computations that lead to your answer. Indicate units of measure.

8 people per hour

(b) Use a trapezoidal sum with three subintervals to estimate the average number of people waiting in line during the first 4 hours that tickets were on sale.

(c) For 0 < t < 9, what is the fewest number of times at which L’(t) must equal 0? Give a reason for your answer.

(d) The rate at which tickets were sold for 0 < t < 9 is modeled by r(t) = 550te – t/2 tickets per hour. Based on the model, how many tickets were sold by 3 P.M. (t = 3), to the nearest whole number?

There were approximately 973 tickets sold by 3 p.m.