An artist makes jewelry from polished stones. The rent for...

An artist makes jewelry from polished stones. The rent for her studio, Internet service, and phone come to $\$ 640$ per month. She also estimates that it costs $\$ 3.50$ in supplies to make one necklace. At art shows and online, she sells the necklaces for $\$ 25$ each. a. Write a linear cost function that represents the $\operatorname{cost} C(x)$ to produce $x$ necklaces during a one-month period. b. Write a linear revenue function to represent the revenue $R(x)$ for selling $x$ necklaces. c. Evaluate $(R-C)(x)$ and interpret its meaning in the context of this problem. d. Determine the profit if the artist sells 212 necklaces during a one-month period.

An artist makes jewelry from polished stones. The rent for her studio, Internet service, and phone come to $\$ 640$ per month. She also estimates that it costs $\$ 3.50$ in supplies to make one necklace. At art shows and online, she sells the necklaces for $\$ 25$ each.
a. Write a linear cost function that represents the $\operatorname{cost} C(x)$ to produce $x$ necklaces during a one-month period.
b. Write a linear revenue function to represent the revenue $R(x)$ for selling $x$ necklaces.
c. Evaluate $(R-C)(x)$ and interpret its meaning in the context of this problem.
d. Determine the profit if the artist sells 212 necklaces during a one-month period.

Key Concepts

Profit Function

The profit function is derived by subtracting the cost function from the revenue function, typically in the form Profit(x) = R(x) - C(x). It represents the net earnings from production and sales, providing insight into the financial viability of a business.

Slope-Intercept Form

The slope-intercept form of a linear equation, expressed as y = mx + b, is commonly used to describe linear relationships. Here, the slope (m) indicates the rate of change (such as cost or revenue per unit), while the intercept (b) represents the initial value (such as fixed costs or starting revenue), making it a valuable tool for modeling economic scenarios.

Linear Revenue Function

A linear revenue function expresses the total income generated from selling products at a fixed price per unit. It is usually represented as R(x) = (price per unit) * x, which illustrates that revenue increases directly with the number of units sold.

Variable Costs

Variable costs change in direct proportion to the level of production. For example, the cost of materials per product is a variable cost that increases as more items are produced, and it must be accounted for when determining the cost to produce each additional unit.

Fixed Costs

Fixed costs are expenses that remain constant regardless of the production level, such as studio rent and subscription fees. They form the baseline of expenses incurred even when no products are produced, and are critical for calculating overall costs in a business model.

Linear Cost Function

A linear cost function models the total production cost as the sum of fixed costs and variable costs that depend on the number of items produced. Typically represented in the form C(x) = fixed cost + (variable cost per unit) * x, it simplifies the process of budgeting and cost analysis.

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