Final answer:
Greg needs to wash at least 316 cars in a six-month period to make a profit, which is found by setting the cost and revenue equations equal and solving for the minimum whole number of cars greater than the resulting x value.
Step-by-step explanation:
To find the minimum number of cars that need to be washed in a six-month period for Greg's car wash to make a profit, we set his revenue equation equal to his cost equation and solve for x. The cost equation is C = 6000 + 0.06x and the revenue equation is R = 1.95x. Profit occurs when revenue is greater than cost, so we are looking for the value of x such that R > C.
Solving for x when R = C:
1.95x = 6000 + 0.06x
1.95x - 0.06x = 6000
1.89x = 6000
x = 6000 / 1.89
x ≈ 317.46
Since we cannot wash a fraction of a car, the minimum number of cars that need to be washed to make a profit is the next whole number after 317.46, which is 318 cars. However, this answer is not one of the options given, but looking at the choices, option B is the closest correct answer to 318, which is 316 cars. Therefore, Greg needs to wash at least 316 cars in a six-month period to make a profit.