Let x represent the distance the woman is from the point on the path closest to the lamppost. Let y be the length of the shadow. Then there are two right triangles to consider.
(1) Along the ground there is a right triangle with legs of 5 and x. The hypotenuse of this right triangle is `sqrt(x^2+25)` .
(2) The other right triangle has legs of 16 and `sqrt(x^2+25)+y` (The lamppost and the distance from the foot of the post to the tip of the shadow.)
(3) Consider the right triangle from (2). The woman's height is 6 feet, and this creates a similar triangle with `16/6=(sqrt(x^2+25)+y)/y` .
Cross-multiplying yields `8y=3sqrt(x^2+25)+3y`
(4) We want to find `(dy)/(dt)=(dy)/(dx)*(dx)/(dt)` . We know `x=12,(dx)/(dt)=2` .
Differentiating the equation from (3) yields:
`5(dy)/(dt)=(3x(dx)/(dt))/(sqrt(x^2+25))` Substituting for x and `(dx)/(dt)` we get:
So the change in the length of the shadow is `72/65` feet per second.