Question

A firm produces a product that has the production cost function C(x)=100x+3125 and the revenue function R(x)=125x. No more than 120 units can be sold. Find and analyze the break-even quantity, then find the profit function.

Answer 1

The student is tasked with finding the break-even point and profit function for a firm's product. The break-even occurs where the cost and revenue functions intersect, and the profit function is the difference between revenue and cost.

Step-by-step explanation:

The student is asking how to calculate the break-even quantity and find the profit function for a firm with a given cost function and a revenue function. To find the break-even quantity, we set the cost function equal to the revenue function and solve for the quantity (x). Given the cost function C(x) = 100x + 3125 and the revenue function R(x) = 125x, setting them equal gives us 125x = 100x + 3125. Solving for x gives us x = 125 units, which is beyond the maximum sales capacity of 120 units, indicating there will be no break-even point within the sales constraints. The profit function is found by subtracting the cost function from the revenue function, which yields P(x) = R(x) - C(x) = 125x - (100x + 3125).