198
3.5. RELATED RATES
both V and r may be viewed as implicit functions of t, with respective derivatives
dV
dt and
dr
dt .
Differentiate both sides of the equation V =
4
3 πr 3 with respect to t (using the chain
rule on the right) to find a formula for
dV
dt that depends on both r and
dr
dt .
(c) At this point in the problem, by differentiating we have “related the rates” of
change of V and r. Recall that we are given in the problem that the balloon is
being inflated at a constant rate of 20 cubic inches per second. Is this rate the
value of
dr
dt or
dV
dt ? Why?
(d) From part (c), we know the value of
dV
dt at every value of t. Next, observe that
when the diameter of the balloon is 12, we know the value of the radius. In the
equation
dV
dt = 4πr 2 dr
dt , substitute these values for the relevant quantities and solve
for the remaining unknown quantity, which is
dr
dt . How fast is the radius changing
at the instant d = 12?
(e) How is the situation different when d = 16? When is the radius changing more
rapidly, when d = 12 or when d = 16?
⊲⊳
Related Rates Problems
In problems where two or more quantities can be related to one another, and all of the
variables involved can be viewed as implicit functions of time, t, we are often interested in
how the rates of change of the individual quantities with respect to time are themselves
related; we call these related rates problems. Often these problems involve identifying one or
more key underlying geometric relationships to relate the variables involved. Once we have
an equation establishing the fundamental relationship among variables, we differentiate
implicitly with respect to time to find connections among the rates of change.
For example, consider the situation where sand is being dumped by a conveyor belt on
a pile so that the sand forms a right circular cone, as pictured in Figure 3.24. As sand falls
from the conveyor belt onto the top of the pile, obviously several features of the sand pile
will change: the volume of the pile will grow, the height will increase, and the radius will
get bigger, too. All of these quantities are related to one another, and the rate at which
each is changing is related to the rate at which sand falls from the conveyor.
The first key steps in any related rates problem involve identifying which variables are
changing and how they are related. In the current problem involving a conical pile of sand,
we observe that the radius and height of the pile are related to the volume of the pile by
the standard equation for the volume of a cone,
V =
1
3
πr
2 h.
3.5. RELATED RATES
both V and r may be viewed as implicit functions of t, with respective derivatives
dV
dt and
dr
dt .
Differentiate both sides of the equation V =
4
3 πr 3 with respect to t (using the chain
rule on the right) to find a formula for
dV
dt that depends on both r and
dr
dt .
(c) At this point in the problem, by differentiating we have “related the rates” of
change of V and r. Recall that we are given in the problem that the balloon is
being inflated at a constant rate of 20 cubic inches per second. Is this rate the
value of
dr
dt or
dV
dt ? Why?
(d) From part (c), we know the value of
dV
dt at every value of t. Next, observe that
when the diameter of the balloon is 12, we know the value of the radius. In the
equation
dV
dt = 4πr 2 dr
dt , substitute these values for the relevant quantities and solve
for the remaining unknown quantity, which is
dr
dt . How fast is the radius changing
at the instant d = 12?
(e) How is the situation different when d = 16? When is the radius changing more
rapidly, when d = 12 or when d = 16?
⊲⊳
Related Rates Problems
In problems where two or more quantities can be related to one another, and all of the
variables involved can be viewed as implicit functions of time, t, we are often interested in
how the rates of change of the individual quantities with respect to time are themselves
related; we call these related rates problems. Often these problems involve identifying one or
more key underlying geometric relationships to relate the variables involved. Once we have
an equation establishing the fundamental relationship among variables, we differentiate
implicitly with respect to time to find connections among the rates of change.
For example, consider the situation where sand is being dumped by a conveyor belt on
a pile so that the sand forms a right circular cone, as pictured in Figure 3.24. As sand falls
from the conveyor belt onto the top of the pile, obviously several features of the sand pile
will change: the volume of the pile will grow, the height will increase, and the radius will
get bigger, too. All of these quantities are related to one another, and the rate at which
each is changing is related to the rate at which sand falls from the conveyor.
The first key steps in any related rates problem involve identifying which variables are
changing and how they are related. In the current problem involving a conical pile of sand,
we observe that the radius and height of the pile are related to the volume of the pile by
the standard equation for the volume of a cone,
V =
1
3
πr
2 h.