Your weekly cost (in dollars) to manufacture x bicycles and...

Your weekly cost (in dollars) to manufacture $x$ bicycles and $y$ tricycles is $$ C(x, y)=24,000+60 x+20 y $$ a. What is the marginal cost of a bicycle? Of a tricycle? HINT [See Example 1.] b. Describe the graph of the cost function $C$. HINT [See Example 7.] c. Describe the slice by $y=100 .$ What cost function does this slice describe? d. Describe the level curve $z=72,000$. What does this curve tell you about costs?

Your weekly cost (in dollars) to manufacture $x$ bicycles and $y$ tricycles is
$$
C(x, y)=24,000+60 x+20 y
$$
a. What is the marginal cost of a bicycle? Of a tricycle? HINT [See Example 1.]
b. Describe the graph of the cost function $C$. HINT [See Example 7.]
c. Describe the slice by $y=100 .$ What cost function does this slice describe?
d. Describe the level curve $z=72,000$. What does this curve tell you about costs?

Key Concepts

Level Curves

Level curves of a multivariable function are the sets of points where the function takes on a constant value. In the context of a cost function, a level curve represents all combinations of products that yield the same total cost. This is useful for determining trade-offs between different products and understanding the financial implications of different production mixes, as the equation of the level curve encapsulates these relationships.

Slicing a Multivariable Function

Slicing a multivariable function involves fixing one of the variables at a constant value to examine the behavior of the function with respect to the remaining variable(s). In a cost function involving multiple products, fixing the quantity of one product produces a single-variable cost function for the other product. This technique simplifies analysis and helps understand the impact of varying one parameter while holding others constant.

Linear Cost Function

A linear cost function is one in which the total cost is expressed as a linear combination of the input quantities plus a fixed cost. Graphically, this represents a plane in three dimensions when considering two products, where the intercept corresponds to fixed costs and the coefficients of the variables are the slopes in their respective directions. This structure facilitates straightforward interpretation of how costs vary with changes in production.

Marginal Cost

Marginal cost refers to the additional cost incurred when producing one extra unit of a good. In the context of a cost function, it is determined by the partial derivative with respect to the respective variable, indicating how cost changes with an incremental increase in production. For example, if the cost function includes a term like 60x, the marginal cost for that product is 60 dollars per unit.

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