Problem 94 - Algebra and Trigonometry

Question

A wholesaler sells graphing calculators. For an order of \(x\) calculators, his total cost in dollars is $$C=50+30 x-0.1 x^{2}$$ and his total revenue is $$R=50 x-0.05 x^{2}$$ (a) Find the profit \(P\) on an order of \(x\) calculators. (b) Find the profit on an order of 10 calculators and on an order of 20 calculators.

Step-by-Step Solution

Answer

Profit function: \(P = 20x + 0.05x^2 - 50\). Profit for 10 calculators is $155, and for 20 calculators is $370.

Step 1: Determine Profit Function

The profit function is defined as the difference between revenue and cost. Therefore, the profit, \( P \), can be calculated as follows:\[P = R - C\]Substituting the given equations,\[P = (50x - 0.05x^2) - (50 + 30x - 0.1x^2)\]

Step 2: Simplify the Profit Function

Simplify the expression for \( P \):\[P = 50x - 0.05x^2 - 50 - 30x + 0.1x^2\]Combine like terms:\[P = (50x - 30x) + (-0.05x^2 + 0.1x^2) - 50\]\[P = 20x + 0.05x^2 - 50\]

Step 3: Calculate Profit for 10 Calculators

Substitute \( x = 10 \) into the profit function:\[P(10) = 20(10) + 0.05(10)^2 - 50\]Calculate:\[P(10) = 200 + 0.05(100) - 50\]\[P(10) = 200 + 5 - 50 = 155\]

Step 4: Calculate Profit for 20 Calculators

Substitute \( x = 20 \) into the profit function:\[P(20) = 20(20) + 0.05(20)^2 - 50\]Calculate:\[P(20) = 400 + 0.05(400) - 50\]\[P(20) = 400 + 20 - 50 = 370\]

Key Concepts

Revenue and Cost Analysis

To understand profit in a business context, we must first differentiate between revenue and cost. Revenue is the total income generated from selling goods or services. In our exercise, where the product is graphing calculators, revenue depends on the quantity sold and the price per calculator, expressed as the formula:

• Revenue = 50x - 0.05x²

Here, the term 50x represents direct income from selling calculators at a base price, while the term -0.05x² accounts for potential discounts or depreciation as more calculators are sold.
On the other hand, cost refers to all expenses incurred in the production and supply of those calculators. This includes fixed costs (constant figures such as rental space), variable costs (increasing with each unit produced), and in this case, diminishing costs at high volumes, expressed as:

• Cost = 50 + 30x - 0.1x²

Fixed costs are represented by 50, while variable costs increase with each additional unit as seen in 30x, and cost reductions at high sales volumes are expressed by -0.1x².
Profit, which is our initial goal to determine, is simply calculated as the difference between total revenue and total cost:

• Profit = Revenue - Cost

Simplifying Expressions

Simplifying algebraic expressions involves rewriting expressions in their simplest form. Often, this means reducing the number of terms and simplifying coefficients while ensuring the operation still represents the original expression.
In this exercise, the given revenue and cost functions, expressed as polynomials, are simplified to derive the profit expression. The profit, measured as the difference between revenue and cost, begins with the equation:

• P = (50x - 0.05x²) - (50 + 30x - 0.1x²)

Breaking it down:

• Subtract each term individually: 50x - 30x, -0.05x² + 0.1x², and introduce -50.
• Combine like terms: this results in 20x from (50x - 30x), 0.05x² from (-0.05x² + 0.1x²), and -50.
• Thus the simplified profit expression becomes:
  • P = 20x + 0.05x² - 50

Simplifying expressions like this makes calculations more straightforward, especially when substituting specific values for further analysis.

Graphing Calculators Financial Mathematics

In financial mathematics, using graphing calculators can help visualize and predict outcomes based on data. These tools make it easier to assess financial equations, like the profit function.
Here's how graphing calculators can boost understanding:

• They create visual plots for functions like the ones derived from profit equations.
• By graphing functions such as: y = 20x + 0.05x² - 50, students can view profit variations over different sales volumes.
• Graphing calculators are excellent for rapid computations like finding a profit for specific quantities, similarly to calculating:
  • P(10) = 20(10) + 0.05(10)² - 50
  • P(20) = 20(20) + 0.05(20)² - 50
  where users input these values to easily find solutions.

By utilizing graphing calculators, financial concepts become more tangible, supporting strategic decision-making processes in businesses.