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3.5. RELATED RATES
Drawing well-labeled diagrams and envisioning how different parts of the figure change is
a key part of understanding related rates problems and being successful at solving them.
Activity 3.14.
A water tank has the shape of an inverted circular cone (point down) with a base of
radius 6 feet and a depth of 8 feet. Suppose that water is being pumped into the tank
at a constant instantaneous rate of 4 cubic feet per minute.
(a) Draw a picture of the conical tank, including a sketch of the water level at a
point in time when the tank is not yet full. Introduce variables that measure
the radius of the water’s surface and the water’s depth in the tank, and label
them on your figure.
(b) Say that r is the radius and h the depth of the water at a given time, t. What
equation relates the radius and height of the water, and why?
(c) Determine an equation that relates the volume of water in the tank at time t to
the depth h of the water at that time.
(d) Through differentiation, find an equation that relates the instantaneous rate
of change of water volume with respect to time to the instantaneous rate of
change of water depth at time t.
(e) Find the instantaneous rate at which the water level is rising when the water in
the tank is 3 feet deep.
(f) When is the water rising most rapidly: at h = 3, h = 4, or h = 5?
⊳
Recognizing familiar geometric configurations is one way that we relate the changing
quantities in a given problem. For instance, while the problem in Activity 3.14 is centered
on a conical tank, one of the most important observations is that there are two key right
triangles present. In another setting, a right triangle might be indicative of an opportunity
to take advantage of the Pythagorean Theorem to relate the legs of the triangle. But in
the conical tank, the fact that the water at any time fills a portion of the tank in such
a way that the ratio of radius to depth is constant turns out to be the most important
relationship with which to work. That enables us to write r in terms of h and reduce the
overall problem to one that involves only one variable, where the volume of water depends
simply on h, and hence to subsequently relate
dV
dt and
dh
dt . In other situations where a
changing angle is involved, a right triangle may offer the opportunity to find relationships
among various parts of the triangle using trigonometric functions.
Activity 3.15.
A television camera is positioned 4000 feet from the base of a rocket launching pad.
The angle of elevation of the camera has to change at the correct rate in order to
keep the rocket in sight. In addition, the auto-focus of the camera has to take into
3.5. RELATED RATES
Drawing well-labeled diagrams and envisioning how different parts of the figure change is
a key part of understanding related rates problems and being successful at solving them.
Activity 3.14.
A water tank has the shape of an inverted circular cone (point down) with a base of
radius 6 feet and a depth of 8 feet. Suppose that water is being pumped into the tank
at a constant instantaneous rate of 4 cubic feet per minute.
(a) Draw a picture of the conical tank, including a sketch of the water level at a
point in time when the tank is not yet full. Introduce variables that measure
the radius of the water’s surface and the water’s depth in the tank, and label
them on your figure.
(b) Say that r is the radius and h the depth of the water at a given time, t. What
equation relates the radius and height of the water, and why?
(c) Determine an equation that relates the volume of water in the tank at time t to
the depth h of the water at that time.
(d) Through differentiation, find an equation that relates the instantaneous rate
of change of water volume with respect to time to the instantaneous rate of
change of water depth at time t.
(e) Find the instantaneous rate at which the water level is rising when the water in
the tank is 3 feet deep.
(f) When is the water rising most rapidly: at h = 3, h = 4, or h = 5?
⊳
Recognizing familiar geometric configurations is one way that we relate the changing
quantities in a given problem. For instance, while the problem in Activity 3.14 is centered
on a conical tank, one of the most important observations is that there are two key right
triangles present. In another setting, a right triangle might be indicative of an opportunity
to take advantage of the Pythagorean Theorem to relate the legs of the triangle. But in
the conical tank, the fact that the water at any time fills a portion of the tank in such
a way that the ratio of radius to depth is constant turns out to be the most important
relationship with which to work. That enables us to write r in terms of h and reduce the
overall problem to one that involves only one variable, where the volume of water depends
simply on h, and hence to subsequently relate
dV
dt and
dh
dt . In other situations where a
changing angle is involved, a right triangle may offer the opportunity to find relationships
among various parts of the triangle using trigonometric functions.
Activity 3.15.
A television camera is positioned 4000 feet from the base of a rocket launching pad.
The angle of elevation of the camera has to change at the correct rate in order to
keep the rocket in sight. In addition, the auto-focus of the camera has to take into