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Understanding profit calculation is crucial when analyzing a business scenario. In this exercise, our aim is to find the number of microwave ovens that must be manufactured to reach a specific profit, in this case, \(1250. Profit can generally be defined as the difference between total revenue and total cost. In this problem, the profit is given as a function of the number of ovens produced, represented by: \[ P = \frac{1}{10} x (300 - x) \] where:

• \( P \) is the profit in dollars,
• \( x \) is the number of ovens produced.

To calculate the profit for any given number of ovens, we substitute that number into the formula. The task is to find the value of \( x \), the number of ovens, that delivers a profit of \\)1250 while staying within the production capacity constraint of 0 to 200 ovens. This involves solving the equation and ensuring that the chosen solution is feasible under the given constraints, emphasizing the importance of checking ranges in real-world problems.

A polynomial expansion is a mathematical process where an expression involving a polynomial is rewritten as a sum of its terms. In the context of our exercise, we are dealing with a product involving terms with variables, and we are required to express it in a standard polynomial format.The profit formula provided is \( \frac{1}{10} x (300 - x) \). To solve the equation effectively, we first remove the fraction by multiplying both sides by 10, producing \( x (300 - x) = 12500 \). Next, we perform polynomial expansion:

• Multiply \( x \) by each term in the expression \( (300-x) \) to obtain:
• \[ 300x - x^2 = 12500 \]

By expanding the expression, the equation is written in the form of a quadratic expression, making it more amenable to solving using algebraic methods like factoring or using the quadratic formula. This process demonstrates how polynomial expansions assist in transforming and simplifying complex algebraic expressions.

Constraint solving is a crucial part of any optimization problem. Here, we are asked to find valid solutions within a specified range, which represents our constraint: the number of microwave ovens manufactured must be between 0 and 200.Once we've expanded and rearranged our equation to the quadratic form, \( x^2 - 300x + 12500 = 0 \), solving it using the quadratic formula gives us two potential values for \( x \): 250 and 50.Here's why constraint solving is essential:

• The result \( x = 250 \) does not satisfy our condition \( 0 \leq x \leq 200 \), so it is discarded.
• The only valid solution within the given constraint is \( x = 50 \).

In mathematics and real-life scenarios alike, it’s vital to ensure that solutions consistently align with predefined constraints. This assures the practicality and relevance of the results obtained from mathematical models.