1. Find two positive numbers whose sum is 300 and whose
product is a maximum. [Solution]
2. Find two positive numbers whose product is 750 and for
which the sum of one and 10 times the other is a minimum. [Solution]
4. We are going to fence in a rectangular field. If we look at the field from above the cost
of the vertical sides are $10/ft, the cost of the bottom is $2/ft and the cost
of the top is $7/ft. If we have $700
determine the dimensions of the field that will maximize the enclosed area. [Solution]
5. We have 45 m^{2} of material to build a box with
a square base and no top. Determine the
dimensions of the box that will maximize the enclosed volume. [Solution]
6. We want to build a box whose base length is 6 times the
base width and the box will enclose 20 in^{3}. The cost of the material of the sides is
$3/in^{2} and the cost of the top and bottom is
$15/in^{2}. Determine the dimensions
of the box that will minimize the cost. [Solution]
7. We want to construct a cylindrical can with a bottom but
no top that will have a volume of 30 cm^{3}. Determine the dimensions of the can that will
minimize the amount of material needed to construct the can. [Solution]
8. We have a piece of cardboard that is 50 cm by 20 cm and
we are going to cut out the corners and fold up the sides to form a box. Determine the height of the box that will
give a maximum volume. [Solution]