How fast is shadow in changing at various points in time.

Formal Write-Up 3
Directions: Read the problem below, carefully. Do not start the problem without first
reading through the rubric. You are responsible for all of the areas on the rubric, even if
you have not read it.
Please attach this to the front of your write-up
This assignment is worth 25 points.
This activity is found in Active Calculus, page 135, Activity 3.13
As pictured in the applet at http:/ a skateboarder who is 6 feet tall rides
under a 15 feet 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. Use Calculus as
you reason through the prompts below.
(a) Draw an appropriate right triangle representation of a snapshot in time of the
skateboarder, once you have seen the applet in the link above. Let � represent the
horizontal distance from the base of the lamppost to the skateboarder and let � represent
the length of his shadow. Make sure your diagram reflects these labeled quantities.
(b) Relate � and �.
(c) Now, consider that � and � are both implicit functions of time. Write an equation that
relates #$


and #&


(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. Use rate of change and/ or Calculus reasoning to justify your response.
(f) Which is moving more rapidly: the skateboarder or the tip of his shadow? Explain,
and justify your answer with evidence from your work.

Sample Solution

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