Exercises
UF Calculus Set 20
1. Suppose we open up the newspaper to find that the average price or gasoline
p
g
is increasing
by 2 cents per day, while the average price of a barrel of oil
p
o
is rising by $
1
.
50
per day.
Assuming all other factors remain constant, at what rate would you say the price of gas is
increasing with respect to the price of oil? Demonstrate this as a related rate, using the fact
that we’ve assumed that
p
g
=
f
(
p
o
)
for some function
f
.
2. A company produces items on an assembly line, and the cost of producing
x
items for the
day is given by
C
(
x
) = 100 + 50
√
x
dollars. It is further known that, due to onsite training,
the daily production
x
of the line will increase. Calculate the rate at which the daily cost of
production changes with time in the instances that follow:
(a) The daily production increases at a constant rate of
0
.
2
items per day. Find the rate of
change of the cost when the daily production is
100
items;
400
items;
900
items.
(b) When the daily production is
x
items, the rate at which the daily production changes is
100
/x
items per day. Find the rate of change of the cost when the daily production is
100
items;
400
items;
900
items.
3. A tank contains 100 grams of a substance dissolved in a large amount of water. The tank is
filtered in such a way that water drains from the tank, leaving the substance behind in the
tank. Consider the volume of the dissolved substance to be negligible. At what rate is the
concentration (grams/liter) of the substance changing with respect to time in each scenario?
(a) the rate after 5 hours, if the tank contains 50 L of water initially, and drains at a constant
rate of 4 L/hr.
(b) the rate at the instant when
20
liters remain, if the water is draining at 2.5 L/hr at that
instant
(c) the rate in scenario (b), if the unknown substance is also being added at a rate of 50g/hr
(and there are 100 grams in the tank at that instant)
4. The seat of a ride at a theme park is conyeyed along the curve
y
= 4 arctan(2
x
)
by a hidden
apparatus that moves the seat in the horizontal direction at a rate of
1
.
5
feet per second. At
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 Spring '08
 ALL
 Calculus, Geometry, Factors, Orders of magnitude

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