Problem 59 - Calculus an Applied Approach

Question

Marginal cost A company manufactures two models of bicycles: a mountain bike and a racing bike. The cost function for producing \(x\) mountain bikes and y racing bikes is given by \(C=10 \sqrt{x y}+149 x+189 y+675\) (a) Find the marginal costs \((\partial C / \partial x \text { and } \partial C / \partial y)\) when \(x=120\) and \(y=160\). (b) When additional production is required, which model of bicycle results in the cost increasing at a higher rate? How can this be determined from the cost model?

Step-by-Step Solution

Answer

The marginal cost for producing mountain bikes is approximately 157, while the marginal cost for producing racing bikes is approximately 192. Therefore, when additional production is required, producing racing bikes results in the cost increasing at a higher rate. These results can be determined from the cost model by calculating and comparing the marginal costs for the two bike models.

Step 1: Compute Marginal Cost

Start by computing the marginal costs, which are the partial derivatives of the cost function with respect to each variable. The derivative with respect to \(x\) is \(\frac{\partial C}{\partial x} = \frac{10y}{2\sqrt{xy}} + 149\), and the derivative with respect to \(y\) is \(\frac{\partial C}{\partial y} = \frac{10x}{2\sqrt{xy}} + 189\)

Step 2: Evaluate Marginal Costs at Given Production Level

Evaluate these derivatives at the given production levels for each type of bike. For mountain bikes, plug \(x = 120\) and \(y = 160\) into the derivative with respect to \(x\) to get \(\frac{\partial C}{\partial x} = 5*160/\sqrt{120*160} + 149 \approx 157\). For racing bikes, plug \(x = 120\) and \(y = 160\) into the derivative with respect to \(y\) to get \(\frac{\partial C}{\partial y} = 5*120/\sqrt{120*160} + 189 \approx 192\)

Step 3: Compare Rates of Cost Increase

Compare the marginal costs for the two bike models. The bike model with the higher marginal cost (\(\frac{\partial C}{\partial y}\) for racing bikes) will result in a higher increase in cost when additional production is required. This is determined from the cost model by noting that the marginal cost is the rate of increase of the cost function with respect to production quantity, and comparing the marginal costs for the two bike models.

Key Concepts

Partial Derivatives

Many real-world problems require understanding how changes in certain variables affect outcomes.
One useful tool for this is partial derivatives, which focus on how a function changes with respect to one variable while keeping other variables constant.
In our exercise, the cost function depends on two variables: the number of mountain bikes, \(x\), and the number of racing bikes, \(y\).
To find the marginal cost for each type of bike, we find the partial derivatives of the cost function.

• The partial derivative with respect to \(x\) looks at how changing the number of mountain bikes affects the overall cost.
• The partial derivative with respect to \(y\) looks at how changing the number of racing bikes affects the overall cost.

By computing these, we determine how sensitive the cost is to producing more of each type of bike, independently of changes in the other type.

Cost Function

A cost function is a mathematical representation of the total cost of production, involving various production factors.
In this exercise, the cost function, \(C=10 \sqrt{x y}+149 x+189 y+675\), incorporates multiple components to express total cost:

• The term \(10 \sqrt{x y}\) represents a shared cost influenced by both types of bikes produced. This could be economies of scale or interaction effects.
• The term \(149 x\) indicates a linear cost associated directly with mountain bikes.
• The term \(189 y\) indicates a linear cost associated with producing racing bikes.
• The constant \(675\) might represent fixed costs that do not change with production levels, like rent or salaries.

The cost function allows businesses to understand their total costs and make decisions on pricing and production strategies accordingly.

Production Levels

Production levels refer to the number of goods being produced in a given scenario.
They play a crucial role in determining the efficiency and cost-effectiveness of manufacturing processes.
In this exercise, you are considering production levels of \(x = 120\) mountain bikes and \(y = 160\) racing bikes.These specific values are used to evaluate the partial derivatives that give the marginal costs:

• For mountain bikes, the marginal cost, \(\frac{\partial C}{\partial x}\), evaluates how the cost changes as more mountain bikes are produced.
• For racing bikes, \(\frac{\partial C}{\partial y}\), measures how the cost increases with additional racing bike production.

Understanding production levels helps in determining where changes in production are most impactful, especially when resources are limited or the market demand fluctuates.