Lesson Find the rate of moving of the tip of a shadow
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<H2>Find the rate of moving of the tip of a shadow</H2> <H3>Problem 1</H3>A street light is at the top of a 18 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip a shadow moving when he is 40 ft from the base of the pole? <B>Solution</B> <pre> In this problem, there are two right-angled triangles: one has a 17 ft tall pole as vertical leg; the other has 6 ft tall man as vertical leg. The hypotenuse is a straight line from the the pole tip to the man' shadow tip. These triangles are similar: they have common acute angle at the tip of the shadow. From similarity, we can write a proportion {{{17/(L+d)}}} = {{{6/d}}}, where L is the current horizontal distance of the man from the pole and d is the length the shadow. From this proportion 17d = 6*(L+d). (1) L and d are functions of time: L = L(t), d = d(t). Let's differentiate (1) over time t. You will get 17*d'(t) = 6*L'(t) + 6*d'(t). (2) The derivative L'(t) is the speed of the man; it is given in the problem, L'(t) = const = 8 ft/s. So, we substitute 8 ft/s instead of L'(t) into (2). We get then 18*d'(t) = 6*8 + 6*d'(t) 18*d'(t) - 6*d'(t) = 48 12*d'(t) = 48 d'(t) = {{{48/12}}} = 4 ft/s. We get an interesting fact: the length of the shadow in this problem, d(t), is a LINEAR function of time with the constant rate of change. The initial length of the shadow at t= 0, d(0), is zero; so, we can write d(t) = 4t feet. The length of the shadow is the linear function d(t) = 4t, in feet. Ok. Let's go further. We are just on the finish line. In this problem, we need to get the derivative over the time of the sum (L+d), or (L(t) + d(t))' = L'(t) + d'(t). As we just noticed above, L'(t) = 8 ft/s, the speed of the man. d'(t) = 4 ft/s; so, the sum is (L(t) + d(t))' = 8 + 4 = 12 ft/s. <U>ANSWER</U>. The tip of the shadow moves at the constant speed of 12 ft/s. </pre> From this problem/solution, we learned an unexpected fact: the tip of the shadow moves at the constant speed, which does not depend on the horizontal distance from the pole. 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