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). 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