A street light is mounted at the top of a 15-ft-tall pole. A man 6 feet tall walks away from the pole with a speed of 7 ft/s along a straight path. How fast (in ft/s) is the tip of his shadow moving when he is 30 feet from the pole? ft/sEnhanced FeedbackPlease try again. Draw a diagram for this problem. Draw a right triangle with horizontal and vertical edges. Let the vertical edge be the pole. Draw another vertical line with one end on the hypotenuse and the other end on the horizontal edge depicting the man. Label the distance between the man and the pole x and the distance between the man and the tip of the shadow y. Use properties of similar triangles to find a relation among x, y, the height of the man, and the height of the pole. Differentiate this equation with respect to time, t, using the Chain Rule, to find the equation for the rate at which the tip of the shadow is moving, ddt(x + y). Then, use the values from the exercise to evaluate the rate of change of the distance between the tip of the shadow and the pole, paying close attention to the signs of the rates of change (positive when increasing, and negative when decreasing).
Question
A street light is mounted at the top of a 15-ft-tall pole. A man 6 feet tall walks away from the pole with a speed of 7 ft/s along a straight path. How fast (in ft/s) is the tip of his shadow moving when he is 30 feet from the pole? ft/sEnhanced FeedbackPlease try again. Draw a diagram for this problem. Draw a right triangle with horizontal and vertical edges. Let the vertical edge be the pole. Draw another vertical line with one end on the hypotenuse and the other end on the horizontal edge depicting the man. Label the distance between the man and the pole x and the distance between the man and the tip of the shadow y. Use properties of similar triangles to find a relation among x, y, the height of the man, and the height of the pole. Differentiate this equation with respect to time, t, using the Chain Rule, to find the equation for the rate at which the tip of the shadow is moving, ddt(x + y). Then, use the values from the exercise to evaluate the rate of change of the distance between the tip of the shadow and the pole, paying close attention to the signs of the rates of change (positive when increasing, and negative when decreasing).
Solution
First, let's draw a diagram and label the distances. We have a right triangle formed by the pole, the ground, and the light ray. The pole is 15 ft tall. The man is 6 ft tall and is walking away from the pole at a speed of 7 ft/s. Let's denote the distance between the man and the pole as x and the length of the shadow as y.
Since the light is at the top of the pole, the tip of the man's shadow creates another right triangle with the man and the light ray. These two triangles are similar because they share an angle and their sides are proportional.
From the similar triangles, we have the following relationship:
15/x = 6/(x+y)
We can rearrange this to get:
15(x+y) = 6x
Then, differentiate both sides with respect to time (t):
15(dx/dt + dy/dt) = 6(dx/dt)
We know that dx/dt = 7 ft/s (the speed of the man), and we want to find dy/dt (the speed of the shadow tip).
Substitute dx/dt = 7 into the equation:
15(7 + dy/dt) = 6*7
Solve for dy/dt:
dy/dt = (67 - 157) / 15 = -7 ft/s
So, the tip of the shadow is moving at a speed of 7 ft/s towards the pole when the man is 30 feet from the pole. The negative sign indicates that the shadow is getting shorter, i.e., the tip of the shadow is moving towards the pole.
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