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# Shadow Problem for AP Calculus

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By McGraw-Hill Professional
Updated on Oct 24, 2011

Practice problems for these concepts can be found at: Applications of Derivatives Practice Problems for AP Calculus

A light on the ground 100 feet from a building is shining at a 6-foot tall man walking away from the streetlight and toward the building at the rate of 4 ft/sec. How fast is his shadow on the building becoming shorter when he is 40 feet from the building? See Figure 8.1-3.

Solution:

Step 1:   Let s be the height of the man's shadow; x be the distance between the man and the light; and t be the time in seconds.

Step 2:   Given: = 4 ft/sec; man is 6 ft tall; distance between light and building =100 ft.

Find .

Step 3:   See Figure 8.1-4. Write an equation using similar triangles, you have:

Step 4:   Differentiate both sides of the equation with respect to t.

Step 5:   Evaluate .

Note: when the man is 40 ft from the building, x (distance from the light) is 60 ft.

Step 6:   The height of the man's shadow on the building is changing at – ft/sec.

Practice problems for these concepts can be found at: Applications of Derivatives Practice Problems for AP Calculus

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