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3.5. RELATED RATES
Provided that
dV
dt is constant, it is immediately apparent that as h gets larger,
dh
dt will get
smaller, while always remaining positive. Hence, the depth of the water is increasing at a
decreasing rate.
Activity 3.16.
As pictured in the applet at http://gvsu.edu/s/9q, a skateboarder who is 6 feet tall
rides under a 15 foot tall lamppost at a constant rate of 3 feet per second. We are
interested in understanding how fast his shadow is changing at various points in time.
(a) Draw an appropriate right triangle that represents a snapshot in time of the
skateboarder, lamppost, and his shadow. Let x denote the horizontal distance
from the base of the lamppost to the skateboarder and s represent the length
of his shadow. Label these quantities, as well as the skateboarder’s height and
the lamppost’s height on the diagram.
(b) Observe that the skateboarder and the lamppost represent parallel line segments
in the diagram, and thus similar triangles are present. Use similar triangles to
establish an equation that relates x and s.
(c) Use your work in (b) to find an equation that relates
dx
dt and
ds
dt .
(d) At what rate is the length of the skateboarder’s shadow increasing at the instant
the skateboarder is 8 feet from the lamppost?
(e) As the skateboarder’s distance from the lamppost increases, is his shadow’s
length increasing at an increasing rate, increasing at a decreasing rate, or
increasing at a constant rate?
(f) Which is moving more rapidly: the skateboarder or the tip of his shadow?
Explain, and justify your answer.
⊳
As we progress further into related rates problems, less direction will be provided. In
the first three activities of this section, we have been provided with guided instruction to
build a solution in a step by step way. For the closing activity and the following exercises,
most of the detailed work is left to the reader.
Activity 3.17.
A baseball diamond is 90 ′ square. A batter hits a ball along the third base line and
runs to first base. At what rate is the distance between the ball and first base changing
when the ball is halfway to third base, if at that instant the ball is traveling 100 feet/sec?
At what rate is the distance between the ball and the runner changing at the same
instant, if at the same instant the runner is 1/8 of the way to first base running at 30
feet/sec?
⊳
3.5. RELATED RATES
Provided that
dV
dt is constant, it is immediately apparent that as h gets larger,
dh
dt will get
smaller, while always remaining positive. Hence, the depth of the water is increasing at a
decreasing rate.
Activity 3.16.
As pictured in the applet at http://gvsu.edu/s/9q, a skateboarder who is 6 feet tall
rides under a 15 foot tall lamppost at a constant rate of 3 feet per second. We are
interested in understanding how fast his shadow is changing at various points in time.
(a) Draw an appropriate right triangle that represents a snapshot in time of the
skateboarder, lamppost, and his shadow. Let x denote the horizontal distance
from the base of the lamppost to the skateboarder and s represent the length
of his shadow. Label these quantities, as well as the skateboarder’s height and
the lamppost’s height on the diagram.
(b) Observe that the skateboarder and the lamppost represent parallel line segments
in the diagram, and thus similar triangles are present. Use similar triangles to
establish an equation that relates x and s.
(c) Use your work in (b) to find an equation that relates
dx
dt and
ds
dt .
(d) At what rate is the length of the skateboarder’s shadow increasing at the instant
the skateboarder is 8 feet from the lamppost?
(e) As the skateboarder’s distance from the lamppost increases, is his shadow’s
length increasing at an increasing rate, increasing at a decreasing rate, or
increasing at a constant rate?
(f) Which is moving more rapidly: the skateboarder or the tip of his shadow?
Explain, and justify your answer.
⊳
As we progress further into related rates problems, less direction will be provided. In
the first three activities of this section, we have been provided with guided instruction to
build a solution in a step by step way. For the closing activity and the following exercises,
most of the detailed work is left to the reader.
Activity 3.17.
A baseball diamond is 90 ′ square. A batter hits a ball along the third base line and
runs to first base. At what rate is the distance between the ball and first base changing
when the ball is halfway to third base, if at that instant the ball is traveling 100 feet/sec?
At what rate is the distance between the ball and the runner changing at the same
instant, if at the same instant the runner is 1/8 of the way to first base running at 30
feet/sec?
⊳