Exercises 33-36 (a) Express the cost C as a function of x,...

Exercises $33-36$ (a) Express the cost $C$ as a function of $x,$ where $x$ represents the number of items as described. (b) Express the revenue $R$ as a function of $x .$ (c) Determine analytically the value of $x$ for which revenue equals cost. (d) Graph $y_{1}=C(x)$ and $y_{2}=R(x)$ on the same $x y$ -axes and interpret the graphs. Delivery Service $A$ truck driver operates a delivery Service. His start-up costs amounted to $\$ 2300 .$ He estimates that it costs him (in terms of gasoline, wear and tear eon his truck, etc.) $\$ 3.00$ per delivery. He charges $\$ 5.50$ per delivery. Let $x$ represent the number of deliveries he makes.

Exercises $33-36$
(a) Express the cost $C$ as a function of $x,$ where $x$ represents the number of items as described.
(b) Express the revenue $R$ as a function of $x .$
(c) Determine analytically the value of $x$ for which revenue equals cost.
(d) Graph $y_{1}=C(x)$ and $y_{2}=R(x)$ on the same $x y$ -axes and interpret the graphs.
Delivery Service $A$ truck driver operates a delivery Service. His start-up costs amounted to $\$ 2300 .$ He estimates that it costs him (in terms of gasoline, wear and tear eon his truck, etc.) $\$ 3.00$ per delivery. He charges $\$ 5.50$ per delivery. Let $x$ represent the number of deliveries he makes.

Key Concepts

Cost Function

The cost function represents the total expense incurred in the production or delivery of services, and is typically composed of fixed costs (which do not vary with the number of items or deliveries) and variable costs (which increase proportionally with the number of items or deliveries). This concept is crucial for understanding how expenses accumulate with increased activity and for making pricing or production decisions.

Revenue Function

The revenue function captures the total income generated from selling goods or providing services. It usually involves multiplying the selling price per unit by the number of units sold. This concept is essential for determining income expectations and for comparing revenue against costs to assess profitability.

Break-even Analysis

Break-even analysis is the process of finding the point at which total revenue equals total cost, meaning that the business neither makes a profit nor incurs a loss. This concept is key in decision-making as it helps in understanding the minimum production or sales volume required to sustain the business, and it informs strategies to manage risks and pricing.

Graphical Analysis

Graphical analysis involves plotting functions such as the cost and revenue functions on the same set of axes to visually compare their behavior over a range of values. This technique not only highlights the break-even point where the graphs intersect, but also provides an intuitive understanding of how variations in production or sales impact profit margins and overall financial performance.

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