A phone manufacturer has a cost of production function 𝐶 of 𝑥 equal to 11𝑥 plus 120 and a revenue function 𝑅 of 𝑥 equal to five 𝑥, where 𝑥 is the number of phones it makes, and 𝐶 of 𝑥 and 𝑅 of 𝑥 are in dollars. How many phones does the company need to manufacture in order to make a profit?
The break-even point for any company occurs when the cost is equal to the revenue. This means that in order to make a profit, the revenue, 𝑅 of 𝑥, must be greater than the cost, 𝐶 of 𝑥. In this question, the revenue or 𝑅 of 𝑥 is equal to five 𝑥. The cost of production is 11𝑥 plus 120. In order to make a profit, five 𝑥 needs to be greater than 11𝑥 plus 120.
We can begin to solve this inequality by subtracting five 𝑥 from both sides. This gives us zero is greater than six 𝑥 plus 120 or six 𝑥 plus 120 is less than zero. We can then subtract 120 from both sides, giving us six 𝑥 is less than negative 120. Finally, we divide both sides of the inequality by six. Dividing a negative number by a positive number gives a negative answer. Therefore, 𝑥 is less than negative 20. 𝑥 is the number of phones that the manufacturer makes. We know that this must be positive. Therefore, 𝑥 cannot be less than negative 20.
We can, therefore, conclude that this company will never be able to make a profit with these current costs of production. This is because the cost, 𝐶 of 𝑥, will always be greater than the revenue, 𝑅 of 𝑥. It doesn’t matter how many phones the company manufactures. They will never make a profit.