Final answer:
To find the number of people the airline would prefer to have on board, we can use the formula x = -b/2a to find the vertex of the profit function. The maximum number of passengers that can board and still allow the airline to profit is 293 or fewer people.
Step-by-step explanation:
To find the number of people that the airline would prefer to have on board, we need to determine the value of x that maximizes the profit. The profit function is given as (1176-4x) rands per person. To maximize the profit, we need to find the value of x that maximizes this function. This corresponds to finding the vertex of the quadratic function. The vertex of a quadratic function in the form of y = ax^2 + bx + c can be found using the formula x = -b/2a.
Substituting the values of a and b from our profit function, we have x = -(-4)/(2(1)), which simplifies to x = 2. Therefore, the airline would prefer to have 2 people on board.
To find the maximum number of passengers that can board such that the airline still profits, we need to determine the range of values for x where the profit is positive. In this case, profit is positive when (1176-4x) > 0. Solving this inequality, we have -4x > -1176, which simplifies to x < 294. Therefore, the maximum number of passengers that can board and still allow the airline to profit is 293 or fewer people.