Related rates problem with a twist

[sebasalekhine7]

I have this problem, and I have attepmted a classical approach without much success.

A man 5 ft tall runs at a rate of 8ft/sec towards a source of light that arises vertically at a point A. The height of the light source H, is given by the formula h(t)=t^3 +1, in feet, where time t is measured in seconds. At what rate is the length of the man's shadow decreasing at time=10seconds?

I have tried establishing the geometrical relationships between the height h(t) and the distance between the point A and the tip of the man's shadow.
say, if we call "s" the length of the shadow, and 'x' the distance between the man and point A, we have that h/5=x/s but then I do not have any other information about the problem. What's next??

 

[HallsofIvy]

Take another look at your picture! At any given instant, we have two triangles: one with a vertical side of length h (the light), the other with a vertical side of length 5 (the man). The horizontal sides of those triangles are s+ x and s respectively. (Notice that there is NO triangle with side length x: there is no triangle with vertex at the man.)

Using "similar triangles", you have s/5= (x+s)/h. Not to convert that from a "static" equation to one involving rates of change, differentiate both sides with respect to t. You are given that dx/dt= -8. Knowing that h(t)= t³+ 1, you can calculate dh/dt, as a function of t, and so find ds/dt as a function of t.

 

[tacman]

It seems like you have made a good start in setting up the problem using the relationships between the height, distance, and length of the shadow. However, in order to solve for the rate of change of the length of the shadow, you will also need to consider the rate of change of the distance between the man and point A. This can be found by taking the derivative of the equation for the distance, which is x=8t.

Using the chain rule, we can then find the derivative of the length of the shadow with respect to time by substituting in the values for x and h(t) and taking the derivative. This will give us an equation for the rate of change of the length of the shadow in terms of time. From there, we can plug in the given time value of t=10 to find the specific rate at that time.

It's important to remember that in related rates problems, we are looking for the rate of change of one variable with respect to time, while holding all other variables constant. By setting up the relationships and using the chain rule, we can solve for this rate of change. Keep practicing and you will get the hang of it!