Question Video: Writing and Solving Linear Equations to Find the Break-Even Point

A phone manufacturer has a cost of production function 𝐢(π‘₯) = 11π‘₯ + 120 and a revenue function 𝑅(π‘₯) = 5π‘₯, where π‘₯ is the number of phones it makes, and 𝐢(π‘₯) and 𝑅(π‘₯) are in dollars. How many phones does the company need to manufacture in order to make a profit?

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Video Transcript

A phone manufacturer has a cost of production function 𝐢 of π‘₯ equal to 11π‘₯ plus 120 and a revenue function 𝑅 of π‘₯ equal to five π‘₯, where π‘₯ is the number of phones it makes, and 𝐢 of π‘₯ and 𝑅 of π‘₯ 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 π‘₯, must be greater than the cost, 𝐢 of π‘₯. In this question, the revenue or 𝑅 of π‘₯ is equal to five π‘₯. The cost of production is 11π‘₯ plus 120. In order to make a profit, five π‘₯ needs to be greater than 11π‘₯ plus 120.

We can begin to solve this inequality by subtracting five π‘₯ from both sides. This gives us zero is greater than six π‘₯ plus 120 or six π‘₯ plus 120 is less than zero. We can then subtract 120 from both sides, giving us six π‘₯ 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, π‘₯ is less than negative 20. π‘₯ is the number of phones that the manufacturer makes. We know that this must be positive. Therefore, π‘₯ 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, 𝐢 of π‘₯, will always be greater than the revenue, 𝑅 of π‘₯. It doesn’t matter how many phones the company manufactures. They will never make a profit.

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