A street light is at the top of a 11.000 ft. tall pole. A man 5.500 ft tall walks away from the pole with a speed of 3.500 feet/sec along a straight path. How fast is the tip of his shadow moving when he is 50.000 feet from the pole?
So, Dx/Dt=3.5, not sure where to go from there.
Best Answer
This is something that you should draw to understand the details. The 11-ft tall pole, the distance from the bottom of the pole to the tip of the man's shadow, and the distance from the top of the pole to the tip of the man's shadow form a large triangle. Additionally, the man's height, the top of the man's head to the tip of his shadow, and the distance from the feet of the man to the tip of the shadow form a smaller triangle. We can use this information to find the desired answer. Indeed, since these triangles described are similar, there is proportionality at play here - we can glean the following equation from examination of the situation, where $h$ is the height of the man's shadow and $x$ is the distance from the bottom of the pole to the man's feet at any given time (thus, $x+h$ is the distance from the bottom of the pole to the tip of the man's shadow - this is the base of the larger triangle), $$\frac{11}{x+h}=\frac{5.5}{h}.$$ Now, rearrange this equation to find a linear relationship between $h$ and $x$: $\,\,\,h=x$. Taking the derivative of both sides of this with respect to $t$, where $t$ is the time in seconds, to get $$\frac{dh}{dt}=\frac{dx}{dt}=3.5$$ It is important to recognize that the speed of the tip of the shadow does not depend upon the distance of the man from the pole, as we can see from the form of $dh/dt$. In other words, the speed of the tip of the shadow is constant at the rate at which the man walks away from the pole, with these dimensions, regardless of how far away he is from the pole.