The cost C (in dollars) of building a house is related to...

The cost $C$ (in dollars) of building a house is related to the number $k$ of carpenters used and the number $e$ of electricians used by $$ C(k, e)=15,000+50 k^{2}+60 e^{2} . $$ If $k=e=10$, compare the effects of increasing $k$ by one unit and of increasing $e$ by one unit. Interpret the result.

The cost $C$ (in dollars) of building a house is related to the number $k$ of carpenters used and the number $e$ of electricians used by
$$
C(k, e)=15,000+50 k^{2}+60 e^{2} .
$$
If $k=e=10$, compare the effects of increasing $k$ by one unit and of increasing $e$ by one unit. Interpret the result.

Key Concepts

Finite Difference Approach

The finite difference approach approximates the change in a function by evaluating the difference in function values after a discrete change in an input variable. This method is particularly useful in determining the exact change in cost when increasing the number of workers by one unit, especially when the function is non-linear.

Partial Derivative

In multivariable calculus, a partial derivative measures the rate at which a function changes as one of its input variables changes while keeping the other variables fixed. In the context of a cost function, partial derivatives are used to assess the marginal impact of increasing one type of labor (such as carpenters or electricians) on the overall cost.

Marginal Cost Analysis

Marginal cost analysis involves computing the additional cost incurred by producing one more unit of a good or service. When applied to labor variables in a cost function, it quantifies the incremental cost of hiring one additional worker, reflecting how the cost increases with each added unit of input.

Quadratic Cost Function

A quadratic cost function includes squared terms of the input variables, which represent non-linear relationships between input levels and total cost. The presence of these squared terms means that the marginal cost increases as the number of workers increases, indicating that each additional worker might be more expensive than the previous one.

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