<p>saints2009, I had my son take a look at the problem, and below is what he had to say. He also mentioned that if you post math problems on [Art</a> of Problem Solving](<a href=“http://www.artofproblemsolving.com%5DArt”>http://www.artofproblemsolving.com), you would most likely get an immediate response as there are a lot of math loving students on the site ready to help others. Good luck.</p>
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<p>Although this has a very messy calculus solution (involving writing the angle in terms of the distance from the observer to the wall ), there is a much nicer solution with no calculus. </p>
<p>Let the wall and the floor intersect at A, let the lower edge of the painting be B, let the upper edge of the painting be C, and suppose the observer stands at point P. Consider the circumcircle w of PBC.</p>
<p>I claim that for the maximum viewing angle, w is tangent to the floor. We use a proof by contradiction. Suppose, for the sake of contradiction, that w is not tangent to the floor; then, w intersects the floor at a point Q not equal to P. Then, take any point R between P and Q. Let line CR intersect w at S (not equal to C). Then, <BSC + <SBR = <BRC ([The</a> Exterior Angle Theorem](<a href=“http://www.cut-the-knot.org/fta/Eat/EAT.shtml]The”>The Exterior Angle Theorem)). Also, <BSC = <BPC ([Inscribed</a> Angles](<a href=“http://www.cut-the-knot.org/Curriculum/Geometry/InscribedAngles.shtml]Inscribed”>Inscribed Angles)), so <BPC + <SBR = <BRC, which means <BRC is greater than <BPC, contradicting the assumption that <BPC is maximal.</p>
<p>Therefore, w is tangent to PBC, so we apply the Power of a Point theorem ([Power</a> of a Point Theorem](<a href=“http://www.cut-the-knot.org/pythagoras/PPower.shtml]Power”>Power of a Point Theorem)) to point A, and we get PA * PA = BA * CA. But BA = 2 (as given in the problem statement) and CA = BA + BC = 2 + 10 = 12, so we get PA * PA = 24. Therefore, PA is the square root of 24, which is our answer (although your teacher might want you to write that as 2 times the square root of 6)</p>
<p>That may have seemed like a long solution, but writing all the calculus out <em>rigorously</em> is much worse.</p>