Final answer:
The break-even point is found by solving the equation set by the cost function equal to the revenue function, 203x + 50 = 10x. However, the calculations result in a negative quantity, suggesting there is no feasible break-even point with the given functions, meaning the cost will always exceed the revenue for any quantity produced.
Step-by-step explanation:
The question asks to determine the break-even point for a company, which occurs when the cost to make items equals the revenue from selling those items. Given the cost function C = 203x + 50 and the revenue function R = 10x, we can find the break-even point by setting C equal to R and solving for x.
To do this, we create the equation 203x + 50 = 10x. Solving for x, we subtract 10x from both sides to get 193x + 50 = 0. Next, we subtract 50 from both sides to get 193x = -50, and finally, we divide both sides by 193 to find x ≈ -0.259. Since the company cannot produce a negative quantity of items, it means there is no break-even point with the given revenue function, because the cost exceeds revenue for every unit produced. However, the information provided in the question might be incorrect, as typically one would expect a positive price for each unit sold. Under the assumption that the revenue per unit is indeed greater than zero, the correct approach is still using the algebraic method described earlier to find the precise quantity where profit is zero.
Now, concerning the revenue at the break-even point, it would be R = 10x when x is the break-even quantity. However, since the calculated x is negative, there would be no revenue at the break-even point because the situation is not economically feasible unless different price data is provided.