A street light is at the top of a 12.5 I . tall pote. A man 5.0 It tall walls away from the pole with a speed of 6.0 feetsec along a straight path. How tast is the tip of his shadow moving when he is 4. feet from the pole? Your answer. Hint: Draw a picture and use simiar thangles

To solve this problem, we can use similar triangles and related rates. Let's denote: - \( h_p = 12.5 \) feet (height of the pole) - \( h_m = 5.0 \) feet (height of the man) - \( x \) = distance from the man to the pole - \( s \) = length of the shadow of the man We know that the man is walking away from the pole at a speed of \( \frac{dx}{dt} = 6.0 \) feet/second. We want to find the speed of the tip of his shadow, which is \( \frac{d(s + x)}{dt} \). ### Step 1: Set up the similar triangles From the pole to the tip of the shadow, we have one triangle, and from the man to the tip of the shadow, we have another triangle. The ratios of the heights to the bases of these triangles are equal: \[ \frac{h_p}{s + x} = \frac{h_m}{s} \] Substituting the heights: \[ \frac{12.5}{s + x} = \frac{5.0}{s} \] ### Step 2: Cross-multiply and simplify Cross-multiplying gives us: \[ 12.5s = 5.0(s + x) \] Expanding the right side: \[ 12.5s = 5.0s + 5.0x \] Rearranging gives: \[ 12.5s - 5.0s = 5.0x \] \[ 7.5s = 5.0x \] ### Step 3: Solve for \( s \) Now, we can express \( s \) in terms of \( x \): \[ s = \frac{5.0}{7.5}x = \frac{2}{3}x \] ### Step 4: Differentiate with respect to time Now we differentiate both sides with respect to time \( t \): \[ \frac{ds}{dt} = \frac{2}{3} \frac{dx}{dt} \] ### Step 5: Substitute known values We know \( \frac{dx}{dt} = 6.0 \) feet/second. Substituting this into the equation gives: \[ \frac{ds}{dt} = \frac{2}{3} \cdot 6.0 = 4.0 \text{ feet/second} \] ### Step 6: Find the speed of the tip of the shadow The speed of the tip of the shadow is the sum of the speed of the man and the speed of the shadow: \[ \frac{d(s + x)}{dt} = \frac{ds}{dt} + \frac{dx}{dt} \] Substituting the values we have: \[ \frac{d(s + x)}{dt} = 4.0 + 6.0 = 10.0 \text{ feet/second} \] ### Conclusion The tip of the man's shadow is moving at a speed of **10.0 feet/second** when he is 4 feet from the pole.