Model For An Airplane Charter Company That Gave Profits

Model For An Airplane Charter Company That Gave The Profitsp Based O

Model for an airplane charter company that gave the profits, P, based on the number, n, of people in a charter group. The equation was: P = -9.8 n^2 + 872 n - 2100 where n is the size of the group that hires the charter company, and P is the total profits the company makes for accepting a charter group of that size. How many people need to be in the group in order for the charter company to have a profit of $9,800?

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The profitability of a charter airline is often directly related to the number of passengers it carries, as larger groups typically generate higher revenues, but also involve higher operational costs. The given profit model, P = -9.8 n^2 + 872 n - 2100, captures this relationship by representing profit (P) as a quadratic function of the number of passengers (n). The negative coefficient of the n^2 term indicates that after a certain point, adding more passengers results in decreasing profit due to increasing costs or operational constraints.

In this analysis, we aim to determine the specific group size, n, that yields a profit of $9,800, as specified in the problem. To do this, we set P = 9,800 and solve for n:

9,800 = -9.8 n^2 + 872 n - 2,100

Rearranging the equation to standard quadratic form gives:

-9.8 n^2 + 872 n - 2,100 - 9,800 = 0

-9.8 n^2 + 872 n - 11,900 = 0

For easier computation, divide the entire equation by -9.8 to normalize the quadratic coefficient:

n^2 - (872 / 9.8) n + (11,900 / 9.8) = 0

Calculating the division yields:

872 / 9.8 ≈ 89.0

11,900 / 9.8 ≈ 1,214.29

So, the quadratic simplifies to:

n^2 - 89.0 n + 1,214.29 = 0

This quadratic can be solved using the quadratic formula, n = [-b ± √(b^2 - 4ac)] / 2a, where a = 1, b = -89.0, and c ≈ 1,214.29.

Calculating the discriminant:

Δ = b^2 - 4ac = (-89.0)^2 - 4 1 1,214.29 = 7,921 - 4,857.16 ≈ 3,063.84

Taking the square root of the discriminant:

√Δ ≈ √3,063.84 ≈ 55.33

Finding the two possible solutions for n:

n = [89.0 ± 55.33] / 2

First solution:

n = (89.0 + 55.33) / 2 ≈ 144.33 / 2 ≈ 72.17

Second solution:

n = (89.0 - 55.33) / 2 ≈ 33.67 / 2 ≈ 16.84

Since the number of passengers n must be a realistic, positive whole number, the solutions approximate to 72 and 17 passengers. These are the group sizes at which the company’s profit reaches approximately $9,800. In practical terms, to achieve at least this profit, the company would need to have a group size close to these values, with the more feasible options being around 17 or 72 passengers, depending on the scenario constraints.

This quadratic analysis illustrates how profit depends on group size and highlights the importance of optimizing passenger numbers for maximum profitability. The model underscores the diminishing returns past a certain point, modeled by the parabola’s vertex, which in this context would occur somewhere between these two solutions.

References

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