Question

The profit earned by the sales division of a company each year can be modeled by the polynomial x - x^2 + 2x - 100, where x is the number of units sold. The profit earned by the manufacturing division can be modeled with the polynomial x^2 - 4x - 300.

A) What is the combined profit of the company in terms of the number of units sold?
B) What is the total profit if the company sells 100 units?
C) Which division is expected to have a higher profit if the company sells 50 units?
D) How many units must the company sell to maximize its total profit?

Answer 1

A) The combined profit of the company in terms of the number of units sold is given by adding the profits of the sales and manufacturing divisions, resulting in the polynomial expression x -
`x^2` + 2x - 100 +
`x^2` - 4x - 300.

B) If the company sells 100 units, substituting x = 100 into the combined profit polynomial yields the total profit earned by the company.

C) To identify the division expected to have a higher profit when 50 units are sold, the individual profits of each division are calculated by substituting x = 50 into their respective polynomial functions.

D) units required to maximize the total profit involves optimizing the combined profit function.

Step-by-step explanation:

Analyzing the given polynomials for the sales and manufacturing divisions, the combined profit is calculated by summing the profit expressions for both divisions. This entails adding the profit function of the sales division, x -
`x^2` + 2x - 100, to the profit function of the manufacturing division,
`x^2` - 4x - 300. This combined expression represents the total profit of the company in terms of units sold.

When the company sells 100 units, substituting this value into the combined profit polynomial allows for the determination of the total profit earned by the company at that sales volume. This involves replacing 'x' in the combined profit function with 100 and solving the expression to find the specific value of the total profit generated.

To identify the division expected to have a higher profit when 50 units are sold, the individual profits of each division are calculated by substituting x = 50 into their respective polynomial functions. Comparing the resultant values will indicate which division is expected to yield a higher profit at this sales volume.

Determining the number of units required to maximize the total profit involves optimizing the combined profit function. This optimization is achieved by finding the maximum value of the combined profit function, typically attained by computing the vertex of the profit equation, which represents the peak point of the profit curve. Calculus techniques or completing the square can be utilized to find the x-value (number of units) corresponding to this maximum profit point.