Do each of the following. (a) Express the cost C as a...

do each of the following. (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 ervice. His start-up costs amounted to $\$ 2300 .$ He estimates that it costs him (in terms of gasoline, wear and tear =on his truck, etc.) $\$ 3.00$ per delivery. He charges $\$ 5.50$ per delivery. Let $x$ represent the number of deliveries he makes.

do each of the following.
(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 ervice. His start-up costs amounted to $\$ 2300 .$ He estimates that it costs him (in terms of gasoline, wear and tear =on his truck, etc.) $\$ 3.00$ per delivery. He charges $\$ 5.50$ per delivery. Let $x$ represent the number of deliveries he makes.

Key Concepts

Linear Functions

Linear functions are mathematical expressions that involve a variable raised to the first power, resulting in a straight-line graph. They are used to model relationships with a constant rate of change, such as cost per item or revenue per delivery, where the total is formed by multiplying the number of items by a rate and then adding any fixed starting value.

Fixed and Variable Costs

Fixed and variable costs are crucial concepts in cost analysis. Fixed costs are constant expenses that do not change with the level of production or services provided, such as start-up fees or base costs, while variable costs depend directly on the quantity produced or delivered. Together, they form the total cost function in a linear model.

Break-even Point Analysis

Break-even analysis involves finding the point where total revenue equals total costs, meaning there is no profit or loss. This analysis is important for determining the minimum number of units or deliveries needed to cover all expenses, and it is computed by setting the cost function equal to the revenue function and solving for the quantity.

Graphical Analysis of Functions

Graphical analysis of functions involves plotting the cost and revenue functions on a coordinate plane to visually identify key points such as the break-even point. By interpreting these graphs, one can assess how the business’s performance changes with the number of items or deliveries and determine the relationships between cost, revenue, and profit.

Transcript

00:01 For number 35, somebody's starting a delivery business, and it costs $2 ,300 to start.

00:09 It costs each $3 for each delivery that they make, and they charge people, customers, $5 .50 for each.

00:19 So we're going to let x represent a delivery.

00:22 We're supposed to come up with a function for cost.

00:26 Well, the cost, of course, depends on how many deliveries they have.

00:29 So the cost is a function of x, and that's going to be, remember, $3 for each delivery, so 3 times x, plus you had that initial startup cost of $2 ,300.

00:47 We'd have an expression for revenue, so of course that also depends on the number of deliveries, and for that they get $5 .50 for each delivery.

00:56 So 5 .5 times x, so times the number of delivers.

01:03 We're to find the break -even point at which they'll start making profit.

01:08 So that's when the cost equals the revenue.

01:12 So i'm just going to set these two equal to each other.

01:15 Think of this as cost is equal to revenue.

01:25 So the cost 3x plus 2300 is equal to 5 .5x.

01:38 I just need to solve for x now.

01:39 So i'm going to subtract 3x from both sides.

01:47 So i get 2 ,300 equals 2 .5.

01:54 X...