Answered

A local park is planning to plant new trees this spring. There will be two types of trees, elm trees and oak trees. Due to environmental constraints, the number of oak trees should be no more than three times the number of elm trees. Oak tree saplings cost $60 each and elm tree saplings cost $80 each. The parks department has $9,600 budgeted for this project.



$\begin{aligned}

60 x+80 y & \leq 9,600 \\

x & \leq 3 y

\end{aligned}$



Which statement represents the given situation?


AThe system represents the minimum amount of money that the parks department can spend on trees and the relationship between the number of oak trees, $x$, and elm trees, $y$.
BThe system represents the minimum amount of money that the parks department can spend on trees and the relationship between the number of elm trees, $x$, and oak trees, $y$.
CThe system represents the maximum amount of money that the parks department can spend on trees and the relationship between the number of elm trees, $x$, and oak trees, $y$.
DThe system represents the maximum amount of money that the parks department can spend on trees and the relationship between the number of oak trees, $x$, and elm trees, $y$.

Verified Answer :

  • The inequality \(60x + 80y \leq 9,600\) models the maximum spending limit based on the cost of oak trees (\(60x\)) and elm trees (\(80y\)).
  • The inequality \(x \leq 3y\) defines the environmental constraint that oak trees (\(x\)) must not exceed three times the count of elm trees (\(y\)).
  • The system correctly maps \(x\) to oak trees and \(y\) to elm trees, identifying the budget as a maximum spending limit.
  • The correct statement is: The system represents the maximum amount of money that the parks department can spend on trees and the relationship between the number of oak trees, \(x\), and elm trees, \(y\). \(\boxed{\text{The system represents the maximum amount of money that the parks department can spend on trees and the relationship between the number of oak trees, } x, \text{ and elm trees, } y.}\)

Explanation

  1. Analyzing the Problem and Given Data

We are given the following system of linear inequalities related to the planting of trees in a park:
\[ \begin{aligned} 60x + 80y &\leq 9,600 \ x &\leq 3y \end{aligned} \]
Here, \(x\) represents the number of oak trees and \(y\) represents the number of elm trees. We are also given the costs for each tree type: oak trees cost \(\$60\) each and elm trees cost \(\$80\) each. The budget for the project is \(\$9,600\). We need to determine the correct interpretation of the system of inequalities given the physical constraints.

  1. Interpreting the Inequalities

The first inequality is \(60x + 80y \leq 9,600\).

  • The term \(60x\) represents the cost of \(x\) oak trees, since each oak tree costs \(\$60\).
  • The term \(80y\) represents the cost of \(y\) elm trees, since each elm tree costs \(\$80\).
  • The sum \(60x + 80y\) is the total cost of all trees planted.
  • The constraint \(60x + 80y \leq 9,600\) states that the total cost must be less than or equal to the total budget of \(\$9,600\). Therefore, this represents the maximum amount of money the parks department can spend on trees.

Next, consider the second inequality \(x \leq 3y\).

  • This inequality states that the number of oak trees (\(x\)) is no more than three times the number of elm trees (\(y\)).
  • This is the environmental constraint imposed on the project.

Thus, \(x\) represents the number of oak trees and \(y\) represents the number of elm trees, and the system sets a cap on the spending based on the budget.

  1. Evaluating Statements and Conclusion

Comparing our findings with the provided options:

  • We determined that the system represents the maximum amount of money the department can spend.
  • We determined that \(x\) represents oak trees and \(y\) represents elm trees.

Checking the given statements:

  1. "The system represents the minimum amount of money..." - Incorrect (we found it represents the maximum).
  2. "The system represents the minimum amount of money..." - Incorrect.
  3. "The system represents the maximum amount... number of elm trees, \(x\), and oak trees, \(y\)." - Incorrect because \(x\) is oak, not elm.
  4. "The system represents the maximum amount of money that the parks department can spend on trees and the relationship between the number of oak trees, \(x\), and elm trees, \(y\)." - This is the correct statement.

Examples

This problem illustrates how businesses and government agencies translate real-world constraints—like budgets and environmental regulations—into mathematical models. By converting narrative constraints into a system of inequalities, planners can find a range of feasible options for resource allocation. For example, if a company wants to distribute its budget across different marketing channels while maintaining a specific ratio of spend between digital and traditional media, they can use a similar system of inequalities to ensure they stay within budget while adhering to their marketing strategy.

Related Questions