A man is 2 m tall is walking at a rate of 1 m per second in a straight line away from a 10 m lamppost. How fast is the tip of his shadow moving away from the lamppost?
If his distance from the pole is x and the length of the shadow is s, then
s/2 = (s+x)/10
1/2 ds/dt = 1/10 (ds/dt + dx/dt)
4 ds/dt = dx/dt
So, the length of the shadow is growing 1/4 as fast as the man's speed.
But that's not how fast the tip of the shadow is moving. You can figure that out, I think.
i am not getting the point,can you elaborate more
To find the speed at which the tip of the man's shadow is moving away from the lamppost, we can use the concept of similar triangles. Let's break down the problem step by step.
Step 1: Determine the length of the man's shadow.
Since the man is 2 meters tall and the lamppost is 10 meters tall, we can conclude that the length of the man's shadow is the same as the distance from the man to the lamppost. Therefore, the length of the man's shadow is also 10 meters.
Step 2: Set up a similar triangle.
We have two similar triangles: one formed by the man, his shadow, and the line connecting them, and another formed by the man, his shadow, and the line connecting the lamppost and the base of the shadow. Let's call the length of the man's shadow "s" and the length of the line connecting the man and the lamppost "x."
Step 3: Apply the concept of similar triangles.
By the properties of similar triangles, the ratio of the lengths of the sides in one triangle is equal to the ratio of the lengths of the corresponding sides in the other. In this case, we have:
(Length of the man's shadow)/(Length of the line connecting the man and the lamppost) = (Height of the man)/(Height of the lamppost)
Substituting the given values, we get:
s/x = 2/10
Step 4: Solve for x.
To find the length of the line connecting the man and the lamppost (x), we'll rearrange the equation:
x = (s * 10)/2
With the length of the man's shadow being 10 meters, we can substitute into the equation:
x = (10 * 10)/2 = 50 meters
Step 5: Find the rate at which the man's shadow is moving away from the lamppost.
Since the man is moving at a rate of 1 meter per second, the rate at which the tip of his shadow is moving away from the lamppost is also 1 meter per second. Therefore, the tip of his shadow is moving away from the lamppost at a rate of 1 meter per second.