Well, well, well! Looks like the light is playing some shadow games with us, huh? Alright, let's shed some light on this situation, shall we?
First, we need to find the length of the man's shadow. Since we have a right triangle formed by the man, his shadow, and the distance from the light, we can use some good ol' Pythagoras.
The hypotenuse of the triangle is the light's height, which is 15 ft. The other leg of the triangle is the length of the man's shadow, and the man's height is the remaining leg.
Using the Pythagorean theorem (a^2 + b^2 = c^2), we can solve for the length of the man's shadow (b):
b^2 = c^2 - a^2
b^2 = 15^2 - 6^2
b^2 = 225 - 36
b^2 = 189
b ≈ 13.75 ft
So, the man's shadow is approximately 13.75 ft long.
Now, to find out how fast the end of the man's shadow is moving when he's walking away from the light at 3 miles per hour, we need to use rates of change, my friend.
Let's say that x represents the distance the man has walked from the light, and y represents the length of the shadow.
Since we know that the man is walking away from the light at a rate of 3 miles per hour, the rate of change of x with respect to time (dx/dt) is 3 mph.
We want to find the rate of change of y with respect to time (dy/dt) when x = some value. In this case, we'll use x = 0 for simplicity.
Using similar triangles, we can set up the following proportion:
x/y = (man's height)/(length of shadow)
0/y = 6/13.75
Cross-multiplying, we get:
0 = 6y/13.75
Now remember, we want to find dy/dt. So, we differentiate both sides of the equation with respect to time.
d(0)/dt = d(6y/13.75)/dt
0 = (6/13.75) * dy/dt
Finally, we can solve for dy/dt:
dy/dt = 0 / (6/13.75)
dy/dt = 0
Voilà ! The end of the man's shadow is not moving when he is walking away from the light at a rate of 3 miles per hour. I guess shadows have a way of sticking around, no matter how fast you walk!