A small business that manufactures specialty phone cases models its weekly profit, P, in dollars, using the equation P…

The given equation, P = 12x - 400, is a linear model for the business's weekly profit. In this equation, x represents the number of phone cases sold. The general form of a linear equation is y = mx + b, where 'b' is the y-intercept, representing the value of y when x = 0. In this context, if the business sells 0 phone cases (x=0), the profit is P = 12(0) - 400 = -400. A profit of -400 dollars is a loss of 400 dollars. This loss, which occurs even with no sales, represents the fixed costs of the business for the week (such as rent, utilities, etc.).

Therefore, the number 400 represents the business's weekly fixed costs.

A is incorrect. This statement invents a relationship between the two numbers in the equation that is not supported by the model. The profit for 12 cases sold would be P = 12(12) - 400 = 144 - 400 = -256 dollars.

B is incorrect. The break-even point is when profit P = 0. Setting P=0 gives 0 = 12x - 400, which means 12x = 400, and x = 400/12 ≈ 33.33.

So, the business needs to sell 34 cases to start making a profit, not 400.

C is the correct answer because the constant term -400 represents the expenses when x=0, which are the fixed costs.

D is incorrect. The profit for each phone case sold is represented by the coefficient of x, which is 12, not 400.