Answer:
2 ft/s
Step-by-step explanation:
The lamppost is 24 ft. tall, and the man is 6 ft. tall. So, we will use a proportion to find the shadow.
Let s is the length of the base of the lamppost to the shadow while x is the length of the base of the lamppost to the man, so the length of the shadow is s - x.
Using triangular ratio, we have;
24/6 = s/(s - x)
4 = s/(s - x)
We cross multiply and distribute to get;
4s - 4x = s
4s - s = 4x
3s = 4x
s = 4x/3
Taking the derivative of both sides according to time, we have;
ds/dt = (4/3)dx/dt
Now, dx/dt is given as 6 ft/s
So;
ds/dt = (4/3) × 6
ds/dt = 8 ft/s
For us to find the rate of length of the shadow according to time, we recall that the shadow = s - x, so we will just take the derivative of each and subtract. Thus;
d(s - x)/dt = ds/dt - dx/dt
Plugging in the relevant values, we have;
ds/dt - dx/dt = 8 - 6 = 2 ft/s
