A photographer is taking a picture of a four-foot painting hung on the art gallery. The camera lens is 1 foot below the lower edge of the painting. How far should the camera be from the painting to maximize the angle subtended by the camera lens?
A photographer is taking a picture of a four-foot painting hung on the art gallery. The camera lens is 1 foot below the lower edge of the painting. How far should the camera be from the painting to maximize the angle subtended by the camera lens?
Let x be the distance the camera is from the painting.
Then, we have two triangles.
Angle $\displaystyle {\theta}$ is the one we need.
So, we have $\displaystyle tan^{-1}(\frac{5}{x})-tan^{-1}(\frac{1}{x})$
Differentiate, set to 0 and solve for x.
We could also use cot:
$\displaystyle cot^{-1}(\frac{x}{5})-cot^{-1}(x)$