Problem 10 - College Algebra and Calculus: An Applied Approach

Question

Write an algebraic expression for the verbal expression. Total Revenue The total revenue obtained by selling \(x\) units at \(\$ 3.59\) per unit

Step-by-Step Solution

Answer

The algebraic expression for the total revenue is \(R = 3.59x\).

Step 1: Understand the Problem

The problem is asking to construct an algebraic expression for the quantity of goods sold (represented by \(x\)) multiplied by the price per unit (which is \(\$ 3.59\)).

Step 2: Formulate the Expression

The algebraic expression for total revenue is obtained by multiplying the number of units, \(x\), by the cost per unit, \(\$ 3.59\). So the total revenue, \(R\), can be expressed as \(R = 3.59 * x\) or \(R = 3.59x\).

Key Concepts

Revenue Calculation

Calculating revenue involves understanding how much money is earned from selling goods or services. In simple terms, revenue is the total amount of money a company receives from sales.
To calculate total revenue, you multiply the number of units sold by the price per unit.
This is crucial because:

• It helps businesses analyze their financial health.
• It provides insight into the company’s growth and sales performance.
• It is foundational for creating profitable pricing strategies.
  

Using algebraic expressions to calculate revenue is efficient because it allows easy modifications.
If either the number of units or the price changes, we can quickly adjust our calculations. For example, if we know the unit price is $3.59 and we sell `x` units, the total revenue can be calculated as:\[R = 3.59 \times x\] This clear mathematical representation aids in accurate forecasting and assessment of sales outcomes.

Unit Price

The unit price is the cost assigned to each single item or unit of product sold. Understanding unit price is essential because it:

• Helps consumers compare the cost-effectiveness of similar products.
• Guides businesses in setting competitive pricing strategies.
• Is vital in determining bulk purchase discounts and promotions.
  

In this exercise, the unit price is given as $3.59. This fixed rate allows us to focus on how changing the number of units sold impacts the total revenue.
By setting a consistent unit price, we can effectively use algebra to predict outcomes based on unit sales. If our price per unit changes due to promotional offers or market shifts, updating our equations to reflect these changes can be swiftly done by changing this single factor.

Multiplication in Algebra

Multiplication is a fundamental operation in algebra, often used to combine variables with constants, like in revenue calculation.
Suppose you sell a product for a certain price per unit. To find out how much money you gain when selling multiple units, you must multiply the price and the number of units. This helps in generalizing a situation where the exact number of units might not be known beforehand.
In algebraic terms, multiplication helps us:

• Create expressions that simplify and represent real-world scenarios.
• Understand relationships between different quantities, such as revenue, pricing, and sales volume.
• Solve for an unknown variable when given certain constraints.
  

In our exercise, the expression \( R = 3.59 \times x \) serves as a simple yet powerful algebraic structure.
It binds the constant price to a variable quantity, making it possible to quickly find the revenue for any number of units.
This flexibility is a key reason algebra is so widely used in business calculations and beyond.