Solve the problem by following the procedure for solving an optimization problem. Show all your steps.

A farmer has 1800 m of fencing to make two equal rectangular pens that share a common side. Determine the dimensions of each pen so that the area is a maximum.

New Horizons Travel advertises a package plan for a Florida vacation. The fare for the flight is $400/person plus $8/person for each unsold seat on the plane. The plane holds 120 passengers and the flight will be cancelled if there are fewer than 50 passengers. What number of passengers will maximize revenue?

An open-topped storage box with a square base has a capacity of 5 m3 . The material for the sides cost $1.60/m2 , while the material for the bottom costs $2.00/ m2 . Find the dimensions that will minimize the cost of making the box.

The perimeter of an isosceles triangle is 38 m. Find the length of the sides of the triangle of maximum area.

An apple orchard now has 80 trees per hectare and the average yield is 400 apples per tree. For each additional tree planted per hectare the average yield per tree is reduced by approximately four apples. How many trees per hectare will give the largest crop of apples?