Country Workshop manufactures both finished and unfinished...

Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week of its rolltop desks in the finished and unfinished versions are $x$ and $y$ units when the corresponding unit prices are $$ \begin{array}{l} p=200-\frac{1}{5} x-\frac{1}{10} y \\ q=160-\frac{1}{10} x-\frac{1}{4} y \end{array} $$ dollars, respectively. a. What is the weekly total revenue function $R(x, y) ?$ b. Find the domain of the function $R$.

Country Workshop manufactures both finished and unfinished furniture for the home. The estimated quantities demanded each week of its rolltop desks in the finished and unfinished versions are $x$ and
$y$ units when the corresponding unit prices are
$$
\begin{array}{l}
p=200-\frac{1}{5} x-\frac{1}{10} y \\
q=160-\frac{1}{10} x-\frac{1}{4} y
\end{array}
$$
dollars, respectively.
a. What is the weekly total revenue function $R(x, y) ?$
b. Find the domain of the function $R$.

Key Concepts

Feasible Region

The feasible region is the set of all points (or combinations of decision variables) that satisfy the problem’s constraints, such as resource limitations, nonnegative conditions, or market restrictions. Determining the feasible region is critical in optimization and analysis, as it identifies the conditions under which the revenue function or any other objective function remains valid.

Domain of a Function

The domain of a function comprises all permissible values of the independent variables for which the function is defined. In the context of revenue functions, the domain is determined by practical and economic constraints, such as ensuring quantities and prices are nonnegative and that any system-imposed limitations are satisfied.

Revenue Function

A revenue function is a mathematical representation used to calculate the total income generated by selling a product or service. It is typically defined as the product of the unit price and the quantity sold. When multiple products or variations are involved, the overall revenue function may be an aggregate of individual revenue contributions, each of which may depend on several variables.

Demand Function

A demand function describes the relationship between the price of a product and the quantity that buyers are willing to purchase. In many cases, including linear models, the function shows how changes in price affect the quantity demanded. This is fundamental in economics to understand consumer behavior and to set pricing strategies.

Functions of Several Variables

When revenue depends on more than one product or decision variable, it is expressed as a function of several variables. This introduces additional complexity, such as the need to evaluate how changes in each independent variable affect the overall outcome. Multivariable functions are essential in modeling scenarios where multiple products or factors interact.

Transcript

Please give Ace some feedback