Question

A landscape architect is ordering trees to plant in a park she is designing. The cost of each maple tree is $40, and the cost of each birch tree is $60. Each maple tree requires 15 gallons of water per week, and each birch tree requires 10 gallons of water per week. The architect has a budget of $2,000 to spend on new trees for the park, and the trees must require no more than 400 gallons of water each week. Determine a system of linear inequalities to represent this situation, where x is the number of maple trees and y is the number of birch trees. Then, select the true statement about the system.

a. The budget constraint is 40x + 60y z 2,000.
b. The maple tree constraint is 40x + 15ys 1,200.
c. The budget constraint is 60x + 40y≤ 2,000.
d. The water constraint is 15x+10y 400.
e. The birch tree constraint is 60x+10y≤ 1,200
f. The water constraint is 10x + 15y = 400.

Answer 1

The correct representations of the system are:

- Budget constraint: \(40x + 60y \leq 2,000\)

- Water constraint: \(15x + 10y \leq 400\)

Options c and f accurately express these constraints.

Let's define the system of linear inequalities based on the given information:

1. Budget Constraint:

The cost of maple trees (\$40 each) and birch trees (\$60 each) must not exceed the budget of \$2,000.

\[ 40x + 60y \leq 2,000 \]

So, option c is correct.

2. Water Constraint:

The total water required by maple trees (15 gallons each) and birch trees (10 gallons each) must not exceed 400 gallons.

\[ 15x + 10y \leq 400 \]

So, option f is correct.

None of the other options accurately represents the system of linear inequalities.