Question

The profit \(P\) for a company producing \(x\) units is \(P=\left(500 x-x^{2}\right)-\left(\frac{1}{2} x^{2}-77 x+3000\right)\) Approximate the change and percent change in profit as production changes from \(x=115\) to \(x=120\) units.

Short answer

The change in profit was \$525 and the percent change in profit was approximately 2.87%.

Step-by-step solution

Simplify the Profit Function

Let's first simplify the profit function. The given profit function is \(P=\left(500x-x^{2}\right)-\left(\frac{1}{2} x^{2}-77 x+3000\right)\). Distributing the subtraction over the terms in the second parentheses gives \(P = 500x - x^2 - \frac{1}{2}x^2 + 77x - 3000\). Combining like terms gives the simplified function \(P = 577x - \frac{3}{2}x^2 - 3000\).

Calculate Initial Profit

Let's subsitute the initial value, \(x=115\), into the profit function to find the initial profit: \(P_{115} = 577(115) - \frac{3}{2}(115)^2 - 3000\). Calculating this gives \(P_{115} = 18315\).

Calculate Final Profit

Next, substitute the final value,\( x=120\), into the profit function to find the final profit: \(P_{120} = 577(120) - \frac{3}{2}(120)^2 - 3000\). Calculating this gives \(P_{120} = 18840\).

Calculate Change in Profit

The change in profit can be found by subtracting the initial profit, \(P_{115}\), from the final profit, \(P_{120}\), which gives: \(\Delta P = P_{120}-P_{115} = 18840 - 18315 = 525\). Thus, the change in profit was \$525.

Calculate Percent Change in Profit

Finally, the percent change in profit can be calculated by dividing the change in profit by the initial profit and multiplying by 100. It's expressed as \(\% \Delta P = \frac{\Delta P}{P_{115}} \times 100 = \frac{525}{18315} \times 100 \approx 2.87\%\). This means the percent change in profit was approximately 2.87%.

Key concepts

Polynomial Functions

Polynomial functions are algebraic expressions that consist of variables raised to whole number powers and are combined using addition, subtraction, and multiplication. They are a fundamental part of algebra and are used to model a wide range of real-world situations. In the context of the profit function given in the exercise, the polynomial function is expressed in terms of the variable \(x\), which represents the number of production units. The function describes how profit changes as the number of units produced changes.

The example in this exercise can be broken down into a polynomial function: \(P = 577x - \frac{3}{2}x^2 - 3000\). Here:

• \(577x\) is a linear term, which means it directly scales with the number of units.
• \(-\frac{3}{2}x^2\) is a quadratic term that represents how increases in production may not result in a proportional increase in profit due to factors like decreasing returns or increasing costs.
• \(-3000\) is a constant term that shifts the function up or down.

Understanding polynomial functions helps in predicting and analyzing scenarios for decision-making purposes in business.

Percent Change

Percent change is a mathematical concept used to express how much a quantity has increased or decreased in percentage terms. This is extremely useful in business, especially when analyzing profit changes, as it gives a relative measure of change compared to the initial amount.

In the context of the given exercise, calculating the percent change in profit involves finding the difference between the final and initial profit, dividing this difference by the initial profit, and then multiplying by 100 to get a percentage. The formula used was:
\(\% \Delta P = \frac{\Delta P}{P_{115}} \times 100\)
Where \(\Delta P\) is the change in profit and \(P_{115}\) represents the initial profit when 115 units are produced. In the example, the percent change was found to be approximately 2.87%, indicating a modest increase."

Production Units

Production units refer to the quantity of items that a company manufactures or produces. In the exercise, the number of production units, denoted by \(x\), plays a crucial role as it directly affects the profit function. The entire context of the problem revolves around changing the number of units from 115 to 120 and observing the resulting effect on profit.

Understanding production units is critical in manufacturing and budgeting processes. Companies use the concept to strategize on scaling production up or down, depending on the costs involved and expected profits. For instance, by calculating profits at different production levels, as done in this exercise, companies can identify the optimal number of units to produce to maximize profits. This involves balancing between producing enough to benefit from economies of scale and avoiding the cost increases associated with higher production levels.

By analyzing the polynomial profit function against varying production units, businesses can make informed decisions on their production strategies.