Activity 3.5.4. 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 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 in- creasing 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.
Activity 3.5.4. 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 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 in- creasing 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.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Transcribed Image Text:Activity 3.5.4. 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 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 in-
creasing 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.
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