With x people on board, a South African airline makes a profit of (1104 − 3 x) rands per person for a specific flight.
A. How many people would the airline prefer to have on board?
Answer: x=
B. What is the maximum number of passengers that can board such that the airline still profits?
Answer: x=
If the profit is (1104-3x) per person, then the total
profit = x(1104-3x) = 1104x - 3x^2
d(profit)/dx = 1104 - 6x
= 0 for max profit
6x = 1104
x = 184 , (max profit = 101568)
To have any profit
x(1104-3x) > 0
critical values are x=0 or x=368
so min number of passengers is 1 for a profit of 1101, max number of passengers is 367 for a profit of 1101
A. How many people would the airline prefer to have on board?
Answer: x = Enough to fill the plane, otherwise it would feel empt(h)y!
B. What is the maximum number of passengers that can board such that the airline still profits?
Answer: x = As many as possible, but it's hard to say because the airline is trying to balance profit and passenger discomfort. They don't want to be overcrowded or hear a lot of whining, so they need to find that perfect sweet spot!
To find the answers, we need to set up the given profit equation and solve it.
Given:
Profit per person = 1104 - 3x rands
Number of people on board = x
A. How many people would the airline prefer to have on board?
To maximize the profit, we would need to find the maximum value of the profit equation (1104 - 3x).
Since the coefficient of x is negative (-3x), the profit equation is a linear equation with a negative slope. This means the profit decreases as the number of people on board increases.
To find the number of people preferred by the airline, we need to find the point where the profit is maximized. This occurs when the slope of the profit equation is zero.
Setting the slope of the profit equation to zero:
-3x = 0
Solving for x:
x = 0
Therefore, the airline would prefer to have 0 people on board (assuming a positive profit is desired). This likely implies that the equation is not applicable for this specific scenario, as we would typically expect the airline to prefer to have at least some passengers on board.
B. What is the maximum number of passengers that can board such that the airline still profits?
To find the maximum number of passengers for which the airline still profits, we need to set the profit equation greater than zero.
Setting the profit equation greater than zero:
1104 - 3x > 0
Solving for x:
-3x > -1104
x < -1104 / -3
x < 368
Therefore, the maximum number of passengers that can board such that the airline still profits is x < 368.
A. To find the number of people the airline would prefer to have on board, we need to determine the maximum profit for the flight. The profit function is given by (1104 - 3x) rands per person, where x represents the number of people on board.
To find the maximum profit, we need to find the value of x that maximizes the profit function. This can be done by finding the derivative of the profit function with respect to x and setting it equal to zero.
Profit function: P(x) = (1104 - 3x) rands per person
Derivative: P'(x) = -3
Setting the derivative equal to zero, we have:
-3 = 0
Since -3 is not equal to zero, there is no x value that maximizes the profit function. This means that the airline does not have a preference for a specific number of people on board in order to maximize profit.
Therefore, the answer to part A is that the airline does not have a specific preference for the number of people on board to maximize profit.
B. The maximum number of passengers that can board such that the airline still profits can be found by setting the profit function greater than zero and solving for x.
Profit function: P(x) = (1104 - 3x) rands per person
Setting the profit function greater than zero, we have:
1104 - 3x > 0
Solving for x, we get:
3x < 1104
x < 368
Therefore, the maximum number of passengers that can board such that the airline still profits is 368.
The answer to part B is x = 368.