Hello, zachb!

I agree with your set-up . . . then I went off on a different path.

A painting in an art gallery has height h and is hung

so that its lower edge is a distance d above the eye of an observer.

How far from the wall should the observer stand to get the best view?

(In other words, where should the observer stand to maximize

the angle theta subtended at his eye by the painting?) Code:

* B
* |
* | h
* |
* * C
* * |
* * | d
* θ * |
A * - - - - - - - - - - - - - - * D
x

We have: .θ .= .arctan[(h+d)/x] - arctan(d/x)

. . which equals: .θ .= .arccot[x/(h+d)] - arccot(x/d)

. . . . . . . . . . . - 1/(h+d) . . . . . 1/d

Then: .θ' .= .----------------- + ------------ .= .0

. . . . . . . . . 1 + [x/(h+d)]² . .1 + (x/d)²

Multiply the first fraction by (h+d)²/(h+d)², the second by d²/d²

. . . . . . . . . . . . . . -(h+d) . . . . . . d

. . and we have: .--------------- + ---------- .= .0

. . . . . . . . . . . . .(h+d)² + x² . . d² + x²

Clear denominators: .-(h+d)(x² + d²) + d[x² + (h+d)²] .= .0

Expand (partially): .-(h+d)x² - d²(h+d) + dx² + d(h+d)² .= .0

. . and we have: .dx² - (h+d)x² .= .d²(h+d) - d(h+d)²

Factor: .[d - (h+d)]·x² .= .d(h+d)·[d - (h+d)]

. . and we have: .-hx² .= .-hd(h+d) . → . x² .= .d(h+d)

. . . . . . . . . . . . . _______

Therefore: .x .= .√d(h + d)

Someone check my work . . . *please!*