Homework 10

Directions: Similar problems may appear on the final. This assignment is not turned in for credit.

  1. Suppose someone owns 4000 meters of fencing. They wish to create a rectangular piece of grazing land where one side is along a river. This means no fence is needed for that side. Moreover, they wish to subdivide the rectangle into three separate sections with two pieces of fence, both of which are parallel to the sides not along the river. What are the dimensions of the largest area that can be enclosed?
  2. By cutting away identical squares from each corner of a rectangular piece of cardboard and folding up the resulting flaps, an open box may be made. If the cardboard is 15 in. long and 8 in. wide, find the dimensions of the box that will yield the maximum volume.
  3. Find the dimensions of a rectangle with a perimeter of 100 ft that has the largest possible area.
  4. Find the absolute extrema, if any, of \[f(x) = \dfrac{1}{2}x^2 - 2\sqrt{x}\qquad [0, 3]\]