Kate's startup birdhouse business has a linear profit model with a slope of 10, meaning she earns $10 in profit per birdhouse sold after costs. The y-intercept is -600, indicating a $600 initial loss. To earn a $100 profit, she must sell at least 70 birdhouses.
The question involves creating a linear function to model the profits of a startup birdhouse business. The function is given as P(x) = 15x - (5x + 600), where x is the number of birdhouses sold and P(x) is the profit.
The total cost is represented by (5x + 600), which includes a fixed startup cost and a variable cost for making each birdhouse.
The y-intercept of the function is -600, meaning that before any sales, the business has incurred a $600 startup cost. This reflects Kate's initial investment needed to start the business.
The slope of the profit function is 10 (15 - 5), not 15 as initially mentioned in the question. This means each birdhouse sold contributes $10 to the profit, after subtracting the cost of making the birdhouse.
To calculate the number of birdhouses Kate needs to sell to earn at least $100 in profit, we set up the inequality 10x - 600 ≥ 100. Solving for x gives us:
10x ≥ 700
x ≥ 70
Therefore, Kate needs to sell at least 70 birdhouses to achieve her profit goal of $100.