Question

A company manufactures two types of wood-burning stoves: a freestanding model and a fireplaceinsert model. The cost function for producing \(x\) freestanding and \(y\) fireplace-insert stoves is \(C=32 \sqrt{x y}+175 x+205 y+1050\) (a) Find the marginal costs \((\partial C / \partial x\) and \(\partial C / \partial y)\) when \(x=80\) and \(y=20\). (b) When additional production is required, which model of stove results in the cost increasing at a higher rate? How can this be determined from the cost model?

Short answer

After evaluating the partial derivatives at (x = 80, y = 20), these values should be compared. The model with the higher value has the higher rate increase in cost for additional production. To determine this solely from the cost model, look at the coefficients of x and y. The model with the larger coefficient will have a higher increase in cost as production increases.

Step-by-step solution

Calculate the Partial Derivatives

To compute the marginal costs, the first thing to do is to calculate the partial derivatives of the cost function \(C\) with respect to \(x\) and \(y\). This can be done using the power rule for derivatives. The partial derivatives are calculated as follows: \[\partial C / \partial x = (1/2) * (32 / \sqrt{x*y}) * y + 175\] and \[\partial C / \partial y = (1/2) * (32 / \sqrt{x*y}) * x + 205\]

Evaluate the Partial Derivatives at Specific Points

Now that the partial derivatives have been found, they can be evaluated at the points (x = 80, y = 20) as specified in the problem. Plug the values into the expressions obtained in step 1 to get: \[\partial C / \partial x |_{x=80, y=20} = (1/2) * (32 / \sqrt{80*20}) * 20 + 175\] and \[\partial C / \partial y |_{x=80, y=20} = (1/2) * (32 / \sqrt{80*20}) * 80 + 205\]

Determine the Model with the Higher Increasing Rate

The marginal costs \(\partial C / \partial x\) and \(\partial C / \partial y\) represent the rate of increase in total production costs if one more stove of each type is produced, respectively. After calculating these values in the previous step, it can be compared to find which one is higher. The model with the higher marginal cost is the one in which cost increases at a higher rate with additional production.

Key concepts

Partial Derivatives

Understanding partial derivatives is essential when dealing with functions of multiple variables, like the cost function for producing stoves in our example.

When we have a function that depends on two or more variables, the rate at which the function changes with respect to one of these variables, while keeping the others constant, is known as a partial derivative. In simpler terms, it shows the sensitivity of the output (in this case, cost) to a change in one input variable (the number of stoves manufactured) when all other variables are held unchanged.

Computed with the same principles as ordinary derivatives, they are imperative for businesses to understand how various factors affect their costs separately. For instance, the partial derivative with respect to variable x gives the additional cost incurred for producing one more unit of the freestanding model, assuming the production of the fireplace-insert model remains constant.

Cost Function

A cost function is a mathematical model that represents the total cost of production as a function of the quantity of goods produced. It encompasses the fixed costs, which do not vary with the level of output, and variable costs, which do.

In our exercise, the cost function is given by the equation C = 32 \( \sqrt{x y} \) + 175x + 205y + 1050, where 'x' and 'y' represent the quantity of freestanding and fireplace-insert stoves produced. Here, 175x and 205y are the variable costs associated with each type of stove, while 32 \( \sqrt{x y} \)\ relates to costs that are interdependent on the production of both types of stoves. Finally, '1050' is the fixed cost, independent of production levels. Understanding this cost function allows the company to predict how changes in production levels affect overall costs.

Being able to break down and analyze cost functions helps businesses in pricing strategies, determining optimal production levels, and budgeting.

Rate of Increase in Production Costs

The rate of increase in production costs refers to how much the total cost increases when the production quantity is increased by one unit. Marginal costs, derived from the partial derivatives, directly tell us about this rate of increase.

By calculating the marginal costs for each product, businesses can determine how much the cost of production would go up if the production of one of the items is slightly increased. This is crucial for management decisions, as it helps in understanding which products are more cost-intensive to produce.

In the given problem, after evaluating the partial derivatives at the production levels of 80 freestanding stove units and 20 fireplace inserts, we deduce which product has a higher marginal cost and consequently shows a higher rate of increase in production costs. This knowledge equips the company with strategic insights for cost management and optimization.