A tree blocks the light coming from a street lamp, casting a shadow. The length

Question

A tree blocks the light coming from a street lamp, casting a shadow. The length l of the shadow is a function of the height h of the lamp. The lamp is attached to a pole, which has a mechanism that changes its height over time. Between the hours of 12 pm and 6 pm, the height of the pole (in feet) is given by the function h = u(t), where t is measured in hours after 12 pm. The table below gives several values of the functions u and u'. Assume the lamp is a horizontal distance of 40 feet from the tree, and that the tree is 25 feet tall. The diagram below depicts this scenario (note that it is not drawn to scale). At 2 pm, what is the length of the shadow? At 4 pm, how fast is the length of the shadow changing? Be sure to consider the meaning of the sign of your answer.

Explanation / Answer

at 2 pm , u = 67 so

25/l = 67/(40+l) from similar triangles

25*40 +25 l = 67 l

l = 500/21 = 23.81 ft -----ANSWER a part

b) first establish relation between height of light and length of shadow

25/l = u/(40+l)

25*40 +25 l = u l

1000 = (u-25) l

l = 1000/(u-25)

dl/dt = -1000/(u-25)^2 *du/dt

at t = 4 , dl/dt = -1000/(57-25)^2 *-6.5 =6.347 -----ANSWER part B

+ve sign implies the length is increasing

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