Problem 84 - Algebra for College Students

Question

Speedy Delivery finds that the cost of making \(x\) deliveries is \(C=3 x+2300,\) while the revenue produced from them is \(R=5.50 x(C\) and \(R\) in dollars).

Step-by-Step Solution

Answer

The profit function is \(P(x) = 2.50x - 2300\) and the break-even point occurs at 920 deliveries.

Step 1: Identify the cost function

The cost function given is: \[C(x) = 3x + 2300\] This represents the total cost in dollars to make \(x\) deliveries.

Step 2: Identify the revenue function

The revenue function given is: \[R(x) = 5.50x\] This represents the total revenue in dollars from making \(x\) deliveries.

Step 3: Set up the profit function

Profit \(P(x)\) is calculated as revenue minus cost: \[P(x) = R(x) - C(x)\] Substitute the given functions: \[P(x) = 5.50x - (3x + 2300)\]

Step 4: Simplify the profit function

Simplify the expression for profit function: \[P(x) = 5.50x - 3x - 2300\] \[P(x) = 2.50x - 2300\]

Step 5: Analyze the break-even point

The break-even point occurs when profit \(P(x)\) is zero (i.e., revenue equals cost). Set \(P(x)\) to zero and solve for \(x\): \[0 = 2.50x - 2300\] \[2.50x = 2300\] \[x = \frac{2300}{2.50}\] \[x = 920\]

Key Concepts

cost function

In economics and business, the cost function represents the total cost of production depending on the level of output or activity, denoted by the variable \(x\). For Speedy Delivery, the cost function is given as \[C(x) = 3x + 2300\]. Here, the term \(3x\) represents the variable costs. These costs change with the number of deliveries made. The constant term \(2300\) is the fixed cost, which remains the same regardless of the number of deliveries.

• Fixed Costs: Costs that do not change with the level of output (e.g., rent, salaries).
• Variable Costs: Costs that vary with the production output (e.g., gasoline, vehicle maintenance).

By understanding the cost function, businesses can predict how costs will increase or decrease based on activity levels.

revenue function

The revenue function represents the total income generated from selling goods or services. For Speedy Delivery, the revenue function is expressed as: \[R(x) = 5.50x\]. This indicates that for every delivery made, the company earns $5.50. The revenue function is straightforward because it is directly proportional to \(x\), the number of deliveries. This means that as deliveries increase, revenue increases proportionally.

• Revenue: The total income from selling goods or services.
• Proportionality: In this case, revenue increases by a fixed amount per unit increase in deliveries.

Understanding the revenue function is crucial for businesses to forecast earnings based on sales or service activity.

profit function

Profit is a key indicator of business success and is calculated as the difference between total revenue and total cost. For Speedy Delivery, the profit function is derived as follows: 1) Start with the revenue function: \[R(x) = 5.50x\]. 2) Subtract the cost function: \[C(x) = 3x + 2300\]. The resulting profit function is: \[P(x) = R(x) - C(x) = 5.50x - (3x + 2300)\]. After simplifying, we get: \[P(x) = 2.50x - 2300\]. This function shows the profit based on the number of deliveries:

• If \(P(x) > 0\), the business is profitable.
• If \(P(x) < 0\), the business is operating at a loss.
• If \(P(x) = 0\), the business is at break-even.

By analyzing the profit function, businesses can determine the profitability at different levels of operation.

break-even point

The break-even point is the level of output where total revenue equals total costs, resulting in a profit of zero. For Speedy Delivery, the break-even point can be calculated using the profit function: \[P(x) = 2.50x - 2300\]. To find the break-even point, set the profit function to zero: \[0 = 2.50x - 2300\]. Solving for \(x\), we get: \[2.50x = 2300\], \[x = \frac{2300}{2.50}\], \[x = 920\]. This means that Speedy Delivery needs to make 920 deliveries to cover all costs and achieve a break-even point. Understanding the break-even point helps businesses make informed decisions regarding production levels and pricing strategies.