Question

Trees in urban areas help keep air fresh by absorbing carbon dioxide. A city has $2100 to spend on planting spruce and maple trees. The land available for planting is 45,000 square feet. Spruce trees cost $30 to plant and require 600 square feet of space. Maple trees cost $40 to plant and require 900 square feet of space. Spruce trees absorb 650 lb/yr of carbon dioxide and maple trees absorb 300 lb/yr of carbon dioxide. How many of each tree should the city plant to maximize carbon dioxide absorption?

Answer 1

Let x =  the number of spruce trees
Let y =  the number of maple trees

The information given is summarized in the following table.
Number  Cost/tree  Area used   CO₂ absorption
-----------   -------------   ---------------  ----------------------
      x         $30           600 ft²         650 lb/yr
      y         $40           900 ft²         300 lb/yr

The amount available to spend is $2100, therefore
30x + 40y ≤ 2100
or
(3/4)x + y ≤ 52.5                (1)

The land available for planting is 45,000 ft², therefore
600x + 900y ≤ 45000
or
(2/3)x + y ≤ 50                    (2)
     
The amount of CO₂ removed per year is
A = 650x + 300y               (3)

The shaded area in the graph shown below is the solution region.
Optimum values of A occur at the vertices, as shown.
The maximum removal rate occurs at (70, 3.33) at a rate of 46,500 lb/year.
Because we should have an integral number of trees, we should have
70 spruce and 3 maple trees.

Answer: 70  spruce, 3 maple