int)
A man of height 1.7 meters walk away from a 5-meter lamppost at a speed of 2.9 m/s. Find the rate at which his shadow is increasing in length.
To find the rate at which the man's shadow is increasing in length, we can use similar triangles.
Let's set up a proportion between the man's height and the length of his shadow:
Height of the man / Length of the man's shadow = Height of the lamppost / Length of the lamppost's shadow
In this case, the height of the man is 1.7 meters, the height of the lamppost is 5 meters, and the length of the lamppost's shadow is fixed at 5 meters.
So, the proportion becomes:
1.7 / x = 5 / 5
Cross-multiplying, we get:
5 * 1.7 = 5 * x
8.5 = 5x
Dividing both sides by 5, we find:
x = 8.5 / 5
x = 1.7
Hence, the initial length of the man's shadow is 1.7 meters.
Now, let's differentiate both sides of the proportion with respect to time to find how the length of the shadow changes over time:
d(Height of the man) / dt / Length of the man's shadow + Height of the man / d(Length of the man's shadow) / dt = 0
We know that d(Height of the man) / dt is the rate at which the man's height is changing, which is 0 since the man's height does not change.
Therefore, the equation becomes:
0 + 1.7 / d(Length of the man's shadow) / dt = 0
Simplifying further:
d(Length of the man's shadow) / dt = -1.7 / 0
Since the rate of change of the man's height is zero, the rate at which his shadow is increasing in length is also zero.
Therefore, the rate at which his shadow is increasing in length is 0 m/s.
make a sketch.
let the length of the man's shadow be x
let the distance between the man and the post be y
by ratios:
1.7/x = 5/(x+y)
1.7x + 1.7y = 5x
1.7y = 3.3x
17y = 33x
17 dy/dt = 33 dx/dt
given: dy/dx = 2.9
33 dx/dt = 17(2.9)
dx/dt = 17(2.9)/33 = appr 1.5 m/s
notice that his shadow is lengthening at a constant rate, and it does not depend on where he is in relation to the post.