Section 4.7: Additional Examples on Optimization  

1. A box with an open top is to be constructed from a square piece of cardboard, 3 ft wide, by cutting out a square from each of the four corners and bending up the sides. Find the largest volume that such a box can have.
(a) Draw diagram(s) illustrating the general situation.
(b) Let \(x\) denote the length of the cut out square, and \(y\) be the length of base. Write an expression for the volume.
(c) Use the given information to write an equation that relates the variables.
(d) Write the volume as a function of one variable.
(e) Finish solving the problem.

2. A farmer wants to fence an area of¬†37.5¬†million square feet in a rectangular field and then divide it in half with a fence parallel to one of the sides of the rectangle. What should the lengths of the sides of the rectangular field be so as to minimize the cost of the fence?    solution

3. A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $20 per linear foot to install and the farmer is not willing to spend more than $6000, find the dimensions for the plot that would enclose the most area.     solution

4. A rectangular storage container with an open top is to have a volume of 10 cubic meters. The length of its base is twice the width. Material for the base costs $8 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.     solution

5. Scientist Sam wants to know how close a comet moving in a parabolic trajectory will get to the sun. We will assume that the sun is located at the origin, the path of the comet follows the parabola \(y = x^2-1\) and that the units on the axes are in millions of miles. The comet is closest to the sun at two points. Find the \(x\)-coordinates of these points and the minimum distance.