Problem 8 - Calculus Concepts

Question

A company receives \(\$ 2.9\) million for each ship it sells and can build the ships for \(\$ 0.2\) million each. a. What is the company's revenue from building and selling a ship? b. Assuming \(\bar{C}(x)\) represents average cost and \(\bar{P}(x)\) represents average profit when \(x\) ships are built and sold, write an expression for the revenue from the building and selling of \(x\) ships.

Step-by-Step Solution

Answer

a. $2.9$ million per ship; b. Revenue expression: $2.9x$.

Step 1: Understanding Revenue per Ship

Revenue is the money that the company earns from selling one ship. According to the problem, the company receives \( \$2.9 \) million for each ship sold.

Step 2: Calculating Cost per Ship

The cost to build one ship is the expense that the company incurs. Each ship costs \( \$0.2 \) million to build.

Step 3: Calculating Revenue from One Ship

The revenue from building and selling one ship is the difference between the selling price and the building cost. This gives us the net amount the company gains per ship.

Step 4: Revenue Expression for Multiple Ships

To find the total revenue for building and selling \( x \) ships, multiply the revenue per ship by \( x \). The revenue expression is \( 2.9x \).

Key Concepts

Average Cost Function

To understand the average cost function, let's break it down. The average cost function, \(ar{C}(x)\), represents the total cost of producing a certain number of goods, divided by the number of goods produced. In simpler terms, it quantifies the cost to produce one unit.

Consider this: if you are making 10 ships and the total cost of making them is \(0.2x\), then to find the average cost per ship, you would divide the total cost by the number of ships. This gives us the formula:

• \(\bar{C}(x) = \frac{0.2x}{x} = 0.2\) million dollars per ship

Now, the key point to remember is:

• The average cost remains constant in this problem, because the cost per ship is constant at \(0.2\) million dollars.

There's no variability in cost as more ships are produced, which can simplify calculations significantly when tackling similar problems.

Average Profit Function

Average profit is a concept that indicates the profit made per unit of product sold. It's derived from subtracting the average cost per unit from the revenue per unit. Let's consider our scenario where \(ar{P}(x)\) represents the average profit function.

To find the average profit per ship, you take the revenue made for each ship (

• \(2.9\) million dollars per ship)

and subtract the average cost per ship (\(0.2\) million dollars). Thus, the equation for average profit per ship would be:

• \(\bar{P}(x) = 2.9 - 0.2 = 2.7\) million dollars per ship.

This means for each ship sold, the company earns \(2.7\) million dollars on average, after deducting the cost to construct each ship.

Revenue Expression

The revenue expression describes how much total revenue the company can expect when it sells multiple ships. When we talk about an expression for revenue, we essentially refer to a formula that calculates the total income from selling any number \(x\) of goods.

Following the problem statement, the revenue for selling a single ship is \(2.9\) million dollars. To calculate the revenue expression for \(x\) number of ships, you multiply the revenue per ship by the number of ships:

• Revenue Expression: \(2.9x\)

This equation demonstrates easily: as the number \(x\) of ships increases, so does the total revenue, linearly. This direct relationship is crucial for businesses to plan production and sales.

Cost Analysis

Analyzing the cost involves understanding all expenses a company incurs to produce a product. In our problem, each ship costs \(0.2\) million dollars to build. Cost analysis helps businesses understand direct costs, which are essential for setting prices and calculating profits.

Some key aspects include:

• **Fixed Costs**: These are constant and do not change with the level of production, like machinery or rental costs. However, in this specific problem, the solution does not highlight any fixed costs.
• **Variable Costs**: These change with production levels. Here, the \(0.2\) million dollar-per-ship cost is a typical example of a variable cost.

By conducting thorough cost analysis, a company can better optimize production strategies and achieve efficient cost management, which directly impacts profit margins.