Problem 35 - A Graphical Approach to College Algebra

Question

Exercises \(33-36\) (a) Express the cost \(C\) as a function of \(x,\) where \(x\) represents the number of items as described. (b) Express the revenue \(R\) as a function of \(x .\) (c) Determine analytically the value of \(x\) for which revenue equals cost. (d) Graph \(y_{1}=C(x)\) and \(y_{2}=R(x)\) on the same \(x y\) -axes and interpret the graphs. Delivery Service \(A\) truck driver operates a delivery Service. His start-up costs amounted to \(\$ 2300 .\) He estimates that it costs him (in terms of gasoline, wear and tear eon his truck, etc.) \(\$ 3.00\) per delivery. He charges \(\$ 5.50\) per delivery. Let \(x\) represent the number of deliveries he makes.

Step-by-Step Solution

Answer

The break-even point occurs at 920 deliveries, where revenue equals cost.

Step 1: Express Cost Function

The cost function, \( C(x) \), includes both the fixed start-up costs and the variable costs per delivery. The start-up cost is given as \( \\(2300 \) and the variable cost per delivery is \( \\)3.00 \). Thus, the total cost can be expressed as: \[ C(x) = 2300 + 3x \] where \( x \) is the number of deliveries.

Step 2: Express Revenue Function

The revenue function, \( R(x) \), is defined by the amount charged per delivery. The driver charges \( \$5.50 \) per delivery. Hence, the revenue function is: \[ R(x) = 5.5x \] where \( x \) is the number of deliveries.

Step 3: Determine Break-Even Point

To find out when revenue equals cost, equate \( C(x) \) to \( R(x) \): \[ 2300 + 3x = 5.5x \]Solve for \( x \):\[ 2300 = 5.5x - 3x \]\[ 2300 = 2.5x \]\[ x = \frac{2300}{2.5} \]\[ x = 920 \]So, the driver needs to make 920 deliveries for the revenue to equal the cost.

Step 4: Graphing and Interpretation

To graph the equations, plot \( y_1 = C(x) = 2300 + 3x \) and \( y_2 = R(x) = 5.5x \) on the same axes. The point at which these two lines intersect corresponds to the break-even point, which is the value of \( x \) where the revenue equals the cost. In this context, when \( x = 920 \), \( y_1 = y_2 \), indicating that both functions have the same value. This point shows where the service starts becoming profitable.

Key Concepts

Cost Function

Understanding the cost function is fundamental in break-even analysis. In our scenario, the cost function, denoted as \( C(x) \), includes all the expenses incurred to deliver packages. It combines fixed and variable costs.

• Fixed cost: This is the initial start-up cost, which is constant regardless of the number of deliveries made. In this case, it's \( \\(2300 \) and remains the same no matter how many deliveries are done.
• Variable cost: This changes based on the number of deliveries. For each delivery, there is an additional cost of \( \\)3.00 \). This is multiplied by the number of deliveries \( x \).

To express this as a function, combine these costs: \[C(x) = 2300 + 3x\] Here, \( C(x) \) represents the total costs when \( x \) deliveries are made. It's a straightforward way to calculate the cumulative expenses by adding the fixed costs to the product of variable costs and number of deliveries.

Revenue Function

The revenue function is equally crucial in understanding a business's financial outcome. It indicates the total income generated from making deliveries. In this context, the revenue function is denoted as \( R(x) \).

• Revenue per unit: It represents the earnings per delivery, which is \( \$5.50 \) in this scenario.
• Total revenue: This is calculated by multiplying the revenue per delivery by the number of deliveries, \( x \).

Thus, the revenue function can be expressed as:\[R(x) = 5.5x\]Here, \( R(x) \) gives the total revenue when \( x \) deliveries are made. It's a simple linear function that helps in forecasting potential earnings based on the number of deliveries.

Graphing Functions

Graphing the cost and revenue functions together is an essential step in visual analysis. By plotting both functions on the same set of axes, you can visually determine the break-even point.

• Cost Graph \( y_1 \): Plot the cost function \( C(x) = 2300 + 3x \). This will appear as a line starting at \( 2300 \) on the y-axis, with a slope of 3, indicating how costs increase with each additional delivery.
• Revenue Graph \( y_2 \): Plot the revenue function \( R(x) = 5.5x \). This line starts at the origin and has a slope of 5.5, showing how revenue rises per delivery.

The intersection point of these lines, where \( y_1 = y_2 \), represents the break-even point. For this problem, it's found at \( x = 920 \), indicating the number of deliveries required where revenue equals cost, showing the transition to profit.

Algebraic Equation Solving

Solving algebraic equations is a critical skill for finding key values, like the break-even point in our exercise. To find where revenue equals cost, set the cost and revenue functions equal to one another:\[2300 + 3x = 5.5x\] First, simplify the equation by subtracting \( 3x \) from both sides:\[2300 = 5.5x - 3x\]This simplifies to:\[2300 = 2.5x\] Next, solve for \( x \) by dividing both sides by 2.5:\[x = \frac{2300}{2.5}\] Calculating this gives:\[x = 920\] So, the driver must complete 920 deliveries to break even. Mastering these algebraic steps is essential for solving many real-world problems involving cost and revenue.