write the constraints as linear inequalities and identify all variables used.
A canoe requires 8 hours of fabrication and a rowboat 5 hours. The fabrication department has at most 110 hours of labor available each week
C=# of canoes to fabricate,
R=# of rowboats to fabricate
C≥0
R≥0
8C+5R≤110
Let's denote the number of canoes as c and the number of rowboats as r. The constraints can be written as linear inequalities as follows:
1. The labor hours used by the canoes must not exceed 110 hours:
8c ≤ 110
2. The labor hours used by the rowboats must not exceed 110 hours:
5r ≤ 110
The variables used are c (number of canoes) and r (number of rowboats).
To write the constraints as linear inequalities and identify the variables used, we can use the following information:
Let's assume that the number of canoes produced is represented by the variable 'x' and the number of rowboats produced is represented by the variable 'y'. The constraints can be formulated as follows:
1. The fabrication time constraint:
- Canoes: Each canoe requires 8 hours of fabrication, so the total fabrication time for x canoes is 8x.
- Rowboats: Each rowboat requires 5 hours of fabrication, so the total fabrication time for y rowboats is 5y.
- Combined, the total fabrication time should be at most 110 hours. Therefore, we can write the first constraint as:
8x + 5y ≤ 110
2. Non-negativity constraint:
- The number of canoes and rowboats cannot be negative, so x ≥ 0 and y ≥ 0.
So, the linear inequalities and variables used are:
- Constraints:
8x + 5y ≤ 110
- Non-negativity:
x ≥ 0
y ≥ 0
Variables Used:
- x: Number of canoes produced
- y: Number of rowboats produced