Question

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 \((\partial C / \partial x\) 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?

Short answer

The marginal costs are approximately 155.16 for the mountain bikes and 194.08 for the racing bikes. Hence, the cost increases at a higher rate for the racing bikes.

Step-by-step solution

Find \(\partial C / \partial x\)

To find the marginal cost with respect to the mountain bikes, we need to differentiate the function \(C(x,y)\) with respect to \(x\). Using the formula for the derivative of a product, we have \(\partial C / \partial x = 10/2*\sqrt{y/x}+149\)

Find \(\partial C / \partial y\)

Now, we need to find the marginal cost with respect to the racing bikes, determine the partial derivative of \(C(x,y)\) with respect to \(y\). Using the formula for the derivative of a product, we have \(\partial C / \partial y = 10/2*\sqrt{x/y}+189\)

Evaluate \(\partial C / \partial x\) and \(\partial C / \partial y\)

Now, we need to evaluate these derivatives at \(x=120\) and \(y=160\). If we plug in these values into the derivative formulas, we find \(\partial C / \partial x = 10/2*\sqrt{160/120}+149 \approx 155.16\) and \(\partial C / \partial y = 10/2*\sqrt{120/160}+189 \approx 194.08\)

Compare the marginal costs

The marginal cost of producing mountain bikes is less than that of producing racing bikes, which means that the cost increases at a higher rate as the production of racing bikes increases compared with the production of mountain bikes. This can be deduced directly from comparing the values of \(\partial C / \partial x\) and \(\partial C / \partial y\)

Key concepts

Partial Derivatives

Understanding partial derivatives is crucial when analyzing functions with multiple variables, such as a cost function in production involving different products. Imagine you have a garden and you're trying to figure out how changing the amount of water affects the growth of tomatoes, while keeping the amount of fertilizer the same. This is similar to what partial derivatives do; they measure the change in a function with respect to one variable while holding the others constant.

In the context of our problem, the cost function involves two types of bicycles, mountain bikes (\(x\)) and racing bikes (\(y\)). To find the marginal costs, which essentially tell us how much the cost will change with the production of one additional unit, we calculate the partial derivatives of the cost function, \( \partial C / \partial x \) for mountain bikes and \( \partial C / \partial y \) for racing bikes. The mathematical operations involve applying derivative rules to the individual variables as if the other variable is a constant. By focusing on one variable at a time, we can analyze how each type of bike influences the overall production cost separately.

Cost Function

The cost function, expressed as \( C = 10 \sqrt{xy} + 149x + 189y + 675 \) in our problem, is essentially a mathematical model that represents the total cost of producing a certain number of goods. Think of it as a recipe that outlines the ingredients (variables like the number of mountain bikes and racing bikes) and the resulting cost (output of the function).

Cost functions are vital for businesses because they help in budgeting and forecasting expenses. The constants and coefficients in the function, such as 149 and 189 in our example, represent fixed and variable costs connected to each product. The first term, \( 10 \sqrt{xy} \) in our equation, could represent a shared resource cost that both types of bikes incur when produced together, reflecting economies of scale in production.

Understanding how each term in the cost function impacts the total cost is essential for managers who make decisions about production levels, pricing, and resource allocation.

Production Optimization

When it comes to production optimization, the goal is to produce goods in the most cost-effective way without compromising on quality. It often involves finding the right mix of products and the quantity to manufacture that leads to the highest profit margins. For this, assessing marginal costs is essential as it tells us at what rate costs are increasing for producing one more unit of each product.

In our example, we computed the marginal costs for mountain bikes and racing bikes. With these calculations, if the company wants to optimize production, they should consider increasing the output of the product with the lower marginal cost when faced with additional demand, as this is generally more cost-efficient. However, this decision might also be subject to other strategic considerations, such as market demand, capacity constraints, and long-term goals. Moreover, understanding the derivatives not only informs us about current cost changes but also provides insights into cost behavior as production scales, which is invaluable for planning and optimization purposes.