How Do You Solve Sphere Inscription and Shadow Length Problems?

[Kobrakai]

I am having trouble with these two problems, I was wondering if anyone here could help me.

1. Given a sphere of radius 10 inches. Calculate the altitude of the inscribed right circular cylinder of maximum volume.

2. A man 6 feet tall walks away from a light 30 feet high at the rate of 3 miles per hour. How fast is the further end of his shadow moving, and how fast is his shadow lengthening?

 

[trancefishy]

this really belongs in the calculus forum.

for the first problem, you can simplify it by looking at only the section of a circle in the first quadrant of the coordinate plane. then look to maximize the area of a rectangle inscribed within the area of the 1/4 circle.

the second problem reminds me that I haven't done a related rates problem in some 6 months, and don't feel like brushing up at the moment :-)

 

[mathwonk]

these are pythagoras and similar triangles problems.