Question

A light on the ground is 30 feet away from a building. A 5 foot tall man is walking from the light to the building at a rate of 3 feet per second. He is casting a shadow on the side of the building. At what rate is his shadow shrinking when he is 10 feet from the building?

Answer 1

Answer:

the shadow is shrinking at 4.5ft/s

Explanation:

Calculing the rate change is a problem that evolves derivates, so you have to find a function that represent the height of the shadow and then derivate that with respect to time.

So first we have to find the function that express the height of the shadow, to do that notice that both the green triangles have the same common blue angle that i pictured in the image, so if we express the tangent function for that angle we have:

$Tan (\alpha)=\frac{5ft(height of the person)}{x(distance of the person from the light)}=\frac{H(Height of the shadow)}{30ft(distance between the ligth and the building)}$

and from here we can tell that:

$H=\frac{5ft*30ft}{x}=\frac{150ft}{x}$

now that we have our function we derivate with respect to time, doing that we have:

$\frac{dH}{dt}=\frac{d}{dt}(\frac{150ft^{2} }{x} )=\frac{-150ft^{2}}{x^{2}}*\frac{dx}{dt}$

where $\frac{dx}{dt}= 3ft/s$

now we just have to replace 10 ft in x and that is the changing rate of the shadow height:

$\frac{dH}{dt}(10ft)=\frac{-150ft^{2}}{(10ft)^{2}}*\frac{3ft}{s}=-4.5ft/s$

and that is it, the minus sign is because the shadow is shrinking instead of expanding.