1. A right triangle has legs of length h and r and a hypotenuse of length 4. It is revolved about the leg of length h to sweep out a circular cone. What values of h and r maximize volume of the cone?
2. A rectangular page in a textbook (with width x and length y) has an area of 98^2, top and bottom margins set at 1 in, and left and right margins set at 1/2 in. The printable area of the page is the rectangle that lies within the margins. What are the dimensions of the page that maximize the printable area?
3. What point on the graph of f(x)= 5/2 - x^2 is closest to the origin? (Hint: you can minimize the square of the distance)
4. A line segment of length 10 joins the points (0,p) and (q,0) to form a triangle in the first quadrant. Find the values of p and q that maximize the area of the triangle.
5. A metal cistern in the shape of a right circular cylinder with volume V= 50 m^3 needs to be painted each year to prevent corrosion. The paint is only applied to surfaces exposed to the elements ( the outside cylinder wall and the circular top.) Find the dimensions r and h of the cylinder that minimize the area of the painted surfaces.