13.9 Day 1 Applications of Extrema

Example 1 A rectangular box is resting on the xy plane with one vertex at the origin. The opposite vertex lies in the plane: 6x + 4y + 3z = 24 Find the maximum volume of such a box (see diagram on page 960)

Example 1 solution Solution: Let x,y and z represent the length, width and height of the box. Because one vertex of the box lines in the plane 6x + 4y + 3z = 24, you know that z = 1/3 (24 – 6x – 4y), and You can write the volume xyz of the box as a function of two variables.

You obtain the crititcal points (0,0) and (4/3,2) You obtain the crititcal points (0,0) and (4/3,2). At (0,0) the volume is 0 so that point does not yield a maximum volume. At the point (4/3,2), you can apply the second partials test.

Example 2 An electronics manufacturer determines that the profit P in dollars is obtained by producing x units of DVD player is approximated by the given model. What is the maximum profit?

Example 2 Solution

Problem 8 Find three positive real numbers x,y,and z such x + y + z = 1 and the sum squares of the three numbers is a minimum.

Problem 8 solution Find critical points and Extrema based on this equation

One day a farmer called up an engineer, a physicist, and a mathematician and asked them to fence off the largest possible area with the least amount of fence. The engineer made the fence in a circle and proclaimed that he had the most efficient design. The physicist made a long, straight line and proclaimed 'We can assume the length is infinite...' and pointed out that fencing off half of the Earth was certainly a more efficient way to do it. The Mathematician just laughed at them. He built a tiny fence around himself and said 'I declare myself to be on the outside.'

Math jokes are the only place where you need a mathematician a physicist and an engineer to all work together to find an area. Niel Chong