Hello, ihmth!

I don't understand the diagram for the first one . . .

2) The lower edge of a picture is ft, the upper edge is ft above the eye of an observer.

At what horizontal distance shall he stand

if the vertical angle subtended by the picture is the greatest? I have a rather unplesant solution . . .

Code:

* P
* | :
* | :
* | :
* | b-a
* | :
* | :
* | :
* * Q
* * | :
* θ * | a
* * | :
E * - - - - - - - - - - - * R
x

is the eye of the observer.

is the height of the picture.

. .

Let , the horizontal distance.

Let

In right triangle

In right triangle

In , use the Law of Cosines: .

We have: .

. . which simplifies to: .

And I let __you__ maximize . . .

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

I have an alternate equation: .

. . but its derivation is even harder to explain.