A man of height 1.4 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.
I agree , except I had 1.127777...
which would round off to 1.128
It will depend on what accuracy you need, or if you are using the concept of significant figures.
To find the rate at which the man's shadow is increasing in length, you need to use similar triangles. Let's denote the length of the man's shadow as "x" and the distance between the man and the lamppost as "y."
Since the triangles formed by the man, his shadow, and the lamppost are similar, we have the following relationship between their corresponding sides:
man's height / man's shadow = distance to lamppost / length of lamppost
In this case, we have:
1.4 / x = y / 5
Rearranging the equation, we have:
x = (1.4/5) * y
To find the rate at which the man's shadow is increasing, we need to differentiate both sides of the equation with respect to time, t:
dx/dt = (1.4 / 5) * dy/dt
Now, we know that the man is walking away from the lamppost at a speed of 2.9 m/s, so dy/dt is equal to -2.9 m/s since the distance is decreasing. Substituting this value in the equation, we get:
dx/dt = (1.4 / 5) * (-2.9)
Now we can calculate the rate at which the man's shadow is increasing:
dx/dt = (1.4 / 5) * (-2.9)
dx/dt ≈ -0.812 m/s
Therefore, the rate at which the man's shadow is increasing is approximately -0.812 m/s. Note that the negative sign indicates that the shadow is decreasing in length, which makes sense because the man is moving away from the lamppost.