Answer:
[tex]\frac{dh}{dt}= -0.9 m/s[/tex]
Step-by-step explanation:
Let h be the length of the shadow. Consider the figure attached. Since the triangles are similar, we have the following relation.
[tex]\frac{2}{h}= \frac{x}{12-x}[/tex]
Which leads to the equation [tex]24-2x=xh[/tex]. Differentiating this equation with respect to x leads to
[tex]-2\frac{dx}{dt}= \frac{dx}{dt}h+\frac{dh}{dt}x[/tex]
We want to find the rate of change for h, then
[tex]\frac{dh}{dt} = -\frac{1}{x}(2\frac{dx}{dt}+\frac{dx}{dt}h)[/tex]
Using the relation we found, we have that [tex]h= \frac{24-2x}{x}[/tex]. Now, we now that the man is 4 m away of the building. That is, that x = 8. Then, h = 1.
So, replacing this value in the equation we have
[tex]\frac{dh}{dt} = -\frac{1}{8}(2\frac{dx}{dt}+\frac{dx}{dt})= -\frac{3}{8}(2.4)= -0.9[/tex]
