In a street, two lamp posts are $300$ ft apart. The light intensity at a distance $d$ from the first lamp post is $1000/d^2$ , the light intensity at distance $d$ from the second (weaker) lamp post is $125/d^2$. In both cases the light intensity is inversely proportional to the square of the distance to the light source).
The $\text{combined light intensity}$ is the sum of the two light intensities coming from both lamp posts.
a) If you are between the lamp posts, at distance $x$ ft from the strong light, then given a formulate for the combined light intensity coming from both lamp posts as a function of $x$
b) Where is the combined light intensity the smallest?
Best Answer
$a) I(x) = \dfrac{1000}{x^2} + \dfrac{125}{(300-x)^2}$
$b) I'(x) = -\dfrac{2000}{x^3}+\dfrac{250}{(300-x)^3} = 0 \iff 8(300-x)^3 = x^3 \iff 2(300-x) = x \iff x = 200$ ft. Thus $I$ is minimized when you are at $200$ ft from the strong intensity lamp post.