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?
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.
