Final answer:
To determine the number of backpacks that can be made each week to achieve a profit of at least $4,000, set up an inequality with the total cost and revenue. Solving the inequality, the number of backpacks is represented by the number line starting from 213 and going to infinity.
Step-by-step explanation:
To determine the number of backpacks that can be made each week to achieve a profit of at least $4,000, we need to consider the total cost and revenue. Let's assume the number of backpacks made each week is represented by 'x'. The total cost to make x backpacks would be the sum of the fixed costs and the variable costs, which is calculated as $1,000 + ($6.45 * x). The total revenue is calculated as the selling price of each backpack multiplied by the number of backpacks made, which is $29.99x.
For the company to make a profit of $4,000, the total revenue needs to be greater than the total cost by at least $4,000. Therefore, we can set up the following inequality:
$29.99x - ($1,000 + ($6.45 * x)) ≥ $4,000
Simplifying the inequality, we get:
$23.54x ≥ $5,000
Dividing both sides of the inequality by $23.54, we find:
x ≥ 212.66
The number of backpacks that can be made each week is represented by the number line starting from 213 and going to infinity, since the company can only make backpacks in multiples of 10.