A man who is 1.6 m tall is walking on the road at a constant speed of 1 m/s. There is (only) one lamp placed 3.2 m above the road. At a specific moment the man was just under the lamp. What will be the rate at which the length of his shadow will be changing after: a) 2 sec b) 5 sec?
To find the rate at which the length of the man's shadow is changing, we can use similar triangles and the concept of related rates.
Let's denote the length of the shadow as "s" (in meters) and the distance between the man and the lamp as "d" (in meters). We know that the man's height is 1.6 m and the height of the lamp is 3.2 m.
We can set up a proportion using the similar triangles formed by the man, his shadow, and the lamp:
d/1.6 = (d+s)/3.2
Now, let's differentiate both sides of the equation with respect to time (t) using implicit differentiation:
d(1.6)/dt = d(d+s)/dt/3.2
The left-hand side represents the rate at which the man's height is changing, which is 0 because he is walking at a constant speed.
Now, let's solve for the rate at which the length of the shadow (s) is changing by plugging in the given values and solving for the unknown quantities:
0 = d(d+s)/dt/3.2
Next, let's rearrange the equation to solve for d(d+s)/dt:
d(d+s)/dt = 0 * 3.2
Since anything multiplied by zero is zero, we have:
d(d+s)/dt = 0
So, the rate at which the length of the man's shadow is changing is 0.
a) After 2 seconds, the rate at which the length of the man's shadow is changing is 0.
b) After 5 seconds, the rate at which the length of the man's shadow is changing is still 0.
Therefore, regardless of the time, the length of the man's shadow will not be changing.