Lesson Find the rate of moving of the tip of a shadow

Find the rate of moving of the tip of a shadow

Problem 1

In this problem, there are two right-angled triangles: one has a 17 ft tall pole as vertical leg;
the other has 6 ft tall man as vertical leg.


The hypotenuse is a straight line from the the pole tip to the man' shadow tip.


These triangles are similar: they have common acute angle at the tip of the shadow.


From similarity, we can write a proportion

     = ,

where L is the current horizontal distance of the man from the pole and d is the length the shadow.

From this proportion

    17d = 6*(L+d).    (1)


L and d are functions of time: L = L(t),  d = d(t). 

Let's differentiate (1) over time t.  You will get

    17*d'(t) = 6*L'(t) + 6*d'(t).   (2)


The derivative L'(t) is the speed of the man; it is given in the problem, L'(t) = const = 8 ft/s.
So, we substitute 8 ft/s instead of L'(t) into (2).  We get then

    18*d'(t) = 6*8 + 6*d'(t)

    18*d'(t) - 6*d'(t)  = 48

    12*d'(t) = 48

    d'(t) =  = 4 ft/s.


We get an interesting fact: the length of the shadow in this problem,  d(t),  is a LINEAR function of time with the constant rate of change.

The initial length of the shadow at t= 0,  d(0),  is zero; so, we can write  d(t) = 4t feet.  

The length of the shadow is the linear function d(t) = 4t, in feet.



           Ok.  Let's go further.  We are just on the finish line.



In this problem, we need to get the derivative over the time of the sum (L+d), or (L(t) + d(t))' = L'(t) + d'(t).


As we just noticed above,  L'(t) = 8 ft/s, the speed of the man.  d'(t) = 4 ft/s;  so, the sum is

    
            (L(t) + d(t))' = 8 + 4 = 12 ft/s.


ANSWER.  The tip of the shadow moves at the constant speed of 12 ft/s.