The city of Normal has $2100 to spend on planting spruce and maple trees. The available land for planting is 45,000 square feet. Spruce trees cost $30 to plant and require 600 square feet of space, absorbing 650 lb/yr of carbon dioxide. Maple trees cost $40 to plant and require 900 square feet of space, absorbing 300 lb/yr of carbon dioxide.

Write a linear programming formulation to decide how many of each type of tree should be planted to maximize carbon dioxide absorption.

Define:
- Let $ x $ be the number of spruce trees planted.
- Let $ y $ be the number of maple trees planted.

Objective function:
Maximize $ 650x + 300y $.

Constraints:
1. Cost constraint: $ 30x + 40y \leq 2100 $
2. Space constraint: $ 600x + 900y \leq 45000 $
3. Non-negativity constraint: $ x \geq 0, y \geq 0 $

Question

25-05-2024
Engineering

College

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The city of Normal has $2100 to spend on planting spruce and maple trees. The available land for planting is 45,000 square feet. Spruce trees cost $30 to plant and require 600 square feet of space, absorbing 650 lb/yr of carbon dioxide. Maple trees cost $40 to plant and require 900 square feet of space, absorbing 300 lb/yr of carbon dioxide.

Write a linear programming formulation to decide how many of each type of tree should be planted to maximize carbon dioxide absorption.

Define:
- Let $ x $ be the number of spruce trees planted.
- Let $ y $ be the number of maple trees planted.

Objective function:
Maximize $ 650x + 300y $.

Constraints:
1. Cost constraint: $ 30x + 40y \leq 2100 $
2. Space constraint: $ 600x + 900y \leq 45000 $
3. Non-negativity constraint: $ x \geq 0, y \geq 0 $

Answer :

Final answer:

The task is to formulate a linear programming problem to maximize carbon dioxide absorption by planting a certain number of spruce and maple trees within a set budget and space limit. By defining variables for the number of each type of tree and setting up an objective function and constraints based on cost and space, we can solve for the optimal tree-planting strategy.

Explanation:

The problem asks to maximize carbon dioxide absorption by determining the number of spruce and maple trees to be planted under budget and space constraints. A linear programming formulation is required to solve the issue. Let x be the number of spruce trees and y be the number of maple trees to plant.

Objective Function:

Maximize Z = 650x + 300y

Subject to Constraints:

Budget: 30x + 40y ≤ 2100 (cost)
Space: 600x + 900y ≤ 45000 (space)

Additionally, we have non-negativity constraints: x ≥ 0 and y ≥ 0 (trees planted cannot be negative).

The objective is to find the values of x and y that maximize Z (carbon dioxide absorption) without violating any of the constraints.

Answer :

Final answer:

Explanation:

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