Problem 96 - Algebra for College Students

Question

Solve each problem. Average profit. The weekly profit in dollars for manufacturing \(x\) bicycles is given by the polynomial \(P(x)=\) \(100 x+2 x^{2} .\) The average profit per bicycle is given by \(A P(x)=\frac{P(x)}{x} .\) Find \(A P(x) .\) Find the average profit per bicycle when 12 bicycles are manufactured.

Step-by-Step Solution

Answer

The average profit per bicycle when 12 bicycles are manufactured is 124 dollars.

Step 1: Understanding the given polynomial

The weekly profit in dollars for manufacturing bicycles is represented by the polynomial \(P(x) = 100x + 2x^2\), where \(x\) is the number of bicycles produced.

Step 2: Writing the formula for average profit

The average profit per bicycle is given by the formula \(AP(x) = \frac{P(x)}{x} \). Substituting for \(P(x)\), we get \[ AP(x) = \frac{100x + 2x^2}{x} \].

Step 3: Simplifying the equation

Simplify the expression by dividing each term in the numerator by \(x\). \[ AP(x) = \frac{100x}{x} + \frac{2x^2}{x} \]. This simplifies to \[ AP(x) = 100 + 2x \].

Step 4: Finding the average profit for 12 bicycles

Substitute \(x = 12\) into the simplified equation. \[ AP(12) = 100 + 2(12) \]. Calculate the result as \[ AP(12) = 100 + 24 = 124 \].

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve a sum of powers of one or more variables multiplied by coefficients. In the exercise, the polynomial given is \(P(x) = 100x + 2x^2\), where \(P(x)\) represents the weekly profit in dollars for manufacturing \(x\) bicycles. Each term in the polynomial adds value depending on the number of bicycles produced. The term \(100x\) shows a linear relationship between profit and bicycles, while \(2x^2\) shows a quadratic relationship, indicating that profit increases faster as more bicycles are produced.
Understanding polynomial functions is crucial, as they help model various real-world scenarios including profit functions. In our case, the polynomial tells us how profit changes with different quantities of output. The degree of the polynomial, which is 2 here, tells us the highest power of the variable in the function and gives us insight into the shape and nature of the profit function.

Average Profit Formula

The average profit per bicycle is a key metric in profitability analysis, providing insight into the efficiency of production. The formula for average profit (\(AP(x)\)) given is \(AP(x) = \frac{P(x)}{x}\), where \(P(x)\) is the total profit for producing \(x\) bicycles. This formula simplifies the understanding of profit per unit, which is critical for assessing the viability of scaling production.

By substituting the polynomial \(P(x) = 100x + 2x^2\) into the average profit formula, we get:
\(AP(x) = \frac{100x + 2x^2}{x}\).
This step sets the foundation for further simplification and helps us find how much profit, on average, each bicycle brings in.

Simplifying Expressions

Simplifying algebraic expressions, especially polynomials, involves reducing them to their simplest form. This makes calculations easier and provides clearer insights. In the average profit calculation, we simplified \(AP(x) = \frac{100x + 2x^2}{x}\) by dividing each term in the numerator by \(x\).

Performing the division for each term, we get:
\( \frac{100x}{x} = 100 \) and \( \frac{2x^2}{x} = 2x \).
Thus, \(AP(x) = 100 + 2x \).
This simple expression not only makes the math easier but also provides a direct relationship between average profit and the number of bicycles produced.

Understanding how to simplify expressions is fundamental, as it transforms complex polynomials into more manageable forms, aiding in both calculations and interpretations.

Substitution in Algebra

Substitution is an efficient tool in algebra, enabling the evaluation of expressions at specific values. In the given exercise, we needed to find the average profit per bicycle for the manufacture of 12 bicycles. We do this by substituting \(x = 12\) into the simplified average profit formula \( AP(x) = 100 + 2x \).

First, apply the substitution:
\( AP(12) = 100 + 2(12) \).
Then, perform the arithmetic:
\( AP(12) = 100 + 24 = 124 \).
This means that the average profit per bicycle when 12 bicycles are manufactured is $124.

Substitution helps in evaluating polynomials and other algebraic expressions quickly, providing specific solutions that are essential for decision-making in real-world scenarios. This straightforward technique transforms abstract formulas into concrete numbers, making it easier to interpret and use the results.