at what distance should you stand from the wall

Question

The lower edge of a 4 foot tall painting is 2 feet above your eye level (see picture below for the idea, but be aware it's not drawn to scale). 

top blue "}" - 4 ft

bottom "}" - 2 ft

At what distance, in feet, should you stand from the wall so your viewing angle of the painting is maximized?

Answer 1

desmos.com/calculator/x61abv4rvi

Answer 2

consider 2 right triangles, one with height 4+2= 6, the other with height 2

base b is the same for both, the distance from observer to the bottom of the wall

the viewing angle -s the inverse tangent of 6/b minus the inverse tangent of 2/b

x = tan^-1(6/b) - tan^-1(2/b) take the derivative, set equal to 0, solve for b. then plug that value of b back into the viewing angle = difference in inverse tangents of 6/b and 2/b.

derivative of inverse tangent of y is 1/sec^2tan^-1(y)

very close to the wall makes the viewing angle nearly zero

very far away makes the viewing angle near zero

somewhere in between is the maximum viewing angle

Just a rough

guestimate but about 3 1/2 looks like the maximum viewing angle of 30 degrees