Answer and Step-by-step explanation:
Let’s solve this problem step by step. First, we can draw a diagram to represent the situation. We have a street light at the top of a 14.0ft tall pole, and a man 5.3ft tall walking away from the pole at a speed of 6.5 feet/sec. We can represent the distance between the man and the pole as x, and the length of the man’s shadow as s.
Since the street light, the top of the man’s head, and the tip of his shadow are collinear, we can use similar triangles to relate x and s. The ratio of the height of the pole to the height of the man is equal to the ratio of x to s, so we have:
(14.0ft)/(5.3ft) = (x + s)/s
Solving for s, we get:
s = (5.3ft * x)/(14.0ft - 5.3ft)
Now, we can take the derivative with respect to time t to find how fast the tip of his shadow is moving:
ds/dt = (5.3ft * dx/dt)/(14.0ft - 5.3ft)
We know that dx/dt is equal to the speed at which the man is walking away from the pole, which is 6.5 feet/sec. Plugging this value into our equation, we get:
ds/dt = (5.3ft * 6.5 feet/sec)/(14.0ft - 5.3ft)
Solving this equation, we find that when he is 43 feet from the pole, the tip of his shadow is moving at a speed of approximately 8.4 feet/sec.