AP Calculus: Chapter 7
Integral Applications
1. (2003 – AB – 2B) A tank contains 125 gallons of heating oil at time t = 0. During the time
interval 0 £ t £12 hours, heating oil is pumped into the tank at the rate of
10
gallons per hour.
H(t) = 2 +
1+ ln ( t +1)
(
)
During the same time interval, heating oil is removed from the tank at the rate
æ t2 ö
R(t) = 12sin ç ÷ gallons per hour.
è 47 ø
a. How many gallons of heating oil are pumped into the tank during the time interval 0 £ t £12
hours?
b. Is the level of heating oil in the tank rising or falling at time t = 6 hours? Give a reason for
your answer.
c. How many gallons of heating oil are in the tank at time t =12 hours?
d. At what time t, for 0 £ t £12 , is the volume of heating oil in the tank the least? Show the
analysis that leads to your conclusion.
(
2. (2008-AB-2B) For time t ³ 0, let r(t) = 120 1- e-10t
2
) represent the speed, in kilometers per hour,
at which a car travels along a straight road. The number of liters of gasoline used by the car to
travel x kilometers is modeled by g(x) = 0.05x 1- e-x 2 .
(
)
a. How many kilometers does the car travel during the first 2 hours?
b. Find the rate of change with respect to time of the number of liters of gasoline used by the car
when t = 2 hours. Indicated units of measure.
c. How many liters of gasoline have been used by the car when it reaches a speed of 80
kilometers per hour?
3. An area was hit by a weekend snowstorm. On Monday morning, the temperature temporarily
rises allowing the snow to melt. However, it is still snowing in the area during the week,
sometimes heavily. At 12:01 am Monday morning, the height of the snow is 5 feet. The height
of the snow changes at a rate modeled by f (t) = sin(t) - 5- t , t days after 12:01am Monday.
a. What is the height of the snow at 12:01am Wednesday morning?
b. If F(t) =
ò
t
0
f (x) dx , using correct units, find and interpret the value of F ¢(2) in terms of the
height of the snow.
c. At what time during the 5-day week was the snow level falling the fastest? Justify your
answer.
d. Write, but do not solve, an equation involving an integral expression whose solution k would
be the number of days the height of the snow would be half of its height at 12:01am Monday.
4. An oil refinery produces oil at a variable rate given by
ì800 if 0 £ t < 30
ï
r(t) = í2600 - 60t if 30 £ t< 40
ï
if t ³ 40
î200
where r(t) is measured in barrels per day.
a. How many barrels are produced in the first 30 days?
b. How many barrels are produced in the first 50 days?
c. Write a piece-wise defined function that gives the number of barrels produced at any time t.
d. Is the rate of production increasing or decreasing at t = 35 days?