Hi Daniel . . I like your name.
This is a tough question. I'm pretty sure I got the answer for (a). I would think that the answer for (b) is the same??
It helps to draw a right triangle with another vertical line inside so you have two similar triangles. The outer vertical line is the lamppost so it's 5 meters long. The inside vertical line is the man so it's 1/5 meters long. The horizontal line is the ground. The length from the bottom of the lamppost to the man is 2 times the elapsed time so we'll call it 2t. If the man has walked one second, it's 2 meters. If he's walked 2 seconds, it's 4 meters . . .
The distance from the man to the corner of the triangle is the length of his shadow. We'll call that X.
Now looking at the similar triangles, the height of the lamppost divided by 2t + X equals the height of the man divided by X:
5/2t+X = 1.5/X
Cross multiply--> 5X = 3t + 1.5X
Subtract 1.5X from both sides--> 3.5X = 3t
Divide both sides by 3.5--> X = .86t
Now find the derivative--> dx/dt = the change in shadow length with respect to time = .86 meters per second
Dan
