A painting in an art gallery has height $h$ and is hung so that lower edge is a distance $d$ above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle $\theta$ subtended at his eye by the painting?)

A) $x = \sqrt { h ( d + h ) }$
B) $x = h ^ { 2 } ( h + d ) ^ { 2 }$
C) $x = d ^ { 2 } ( h + d ) ^ { 2 }$
D) $x = \sqrt { d ( h + d ) }$
E) $x = h ^ { 2 } ( h - d ) ^ { 2 }$

Question

A painting in an art gallery has height $h$ and is hung so that lower edge is a distance $d$ above the eye of an observer (as in the figure). How far from the wall should the observer stand to get the best view? (In other words, where should the observer stand so as to maximize the angle $\theta$ subtended at his eye by the painting?)

A) $x = \sqrt { h ( d + h ) }$ 
B) $x = h ^ { 2 } ( h + d ) ^ { 2 }$ 
C) $x = d ^ { 2 } ( h + d ) ^ { 2 }$ 
D) $x = \sqrt { d ( h + d ) }$ 
E) $x = h ^ { 2 } ( h - d ) ^ { 2 }$

Quizplus · Accepted Answer

The angle $\theta$ subtended by the painting at the observer's eye is given by $\tan \theta = \frac { h } { d + x }$. To maximize this angle, we need to maximize $\frac { h } { d + x }$, which means we need to minimize the denominator. Therefore, we should choose the option that has the square root of a product of factors involving $d+h$ and/or $h$, since this will minimize the sum of the denominators inside the square roots. The only option that fits this description is D, which gives $x = \sqrt{d(h+d)}$.

A Painting in an Art Gallery Has Height hhh And Is Hung So That Lower Edge Is a Distance

A Painting in an Art Gallery Has Height $h$ And Is Hung So That Lower Edge Is a Distance