A firm will break even (no profit and no loss as long as...

A firm will break even (no profit and no loss as long as revenue just equals cost. The value of $x$ (the number of items produced and sold where $C(x)=R(x)$ is called the break-even point. Assume that each of the following can be expressed as a linear function. Find (a) the cost function, (b) the revenue function, and $(c)$ the profit function. (d) Find the break-even point and decide whether the product should be produced based on the restrictions on sales. $$\begin{array}{c|c|c}\text { Fixed cost } & \text { Variable cost } & \text { Price of Item } \\ \hline \$ 2700 & \$ 150 & \$ 280 \\\hline\end{array}$$ No more than 25 units can be sold.

A firm will break even (no profit and no loss as long as revenue just equals cost. The value of $x$ (the number of items produced and sold where $C(x)=R(x)$ is called the break-even point. Assume that each of the following can be expressed as a linear function. Find
(a) the cost function,
(b) the revenue function, and $(c)$ the profit function.
(d) Find the break-even point and decide whether the product should be produced based
on the restrictions on sales.
$$\begin{array}{c|c|c}\text { Fixed cost } & \text { Variable cost } & \text { Price of Item } \\
\hline \$ 2700 & \$ 150 & \$ 280 \\\hline\end{array}$$ No more than 25 units can be sold.

Key Concepts

Linear Functions

Linear functions are mathematical representations where the relationship between variables is expressed as a line with a constant slope and an intercept. In business contexts, this simplifies the relationships between costs, revenues, and profits, making it easier to predict changes when the number of units produced or sold varies.

Cost Function

The cost function represents the total expenses incurred by a firm in the production process. It is typically composed of fixed costs, which remain constant regardless of the production level, and variable costs, which change in proportion to the number of units produced. The cost function is crucial for understanding how production levels impact overall expenditure.

Revenue Function

The revenue function calculates the total income generated from the sale of goods or services. It is computed by multiplying the number of items sold by the unit selling price. Understanding the revenue function helps in determining the income side of the profit equation and in planning sales strategies.

Profit Function

The profit function is derived by subtracting the total cost from the total revenue, providing a measure of the financial gain or loss from a business activity. It serves as a critical metric for assessing business performance and making decisions related to production, pricing, and market strategies.

Break-even Analysis

Break-even analysis involves determining the point at which total revenue equals total cost, which is known as the break-even point. At this point, a firm neither makes a profit nor incurs a loss. This analysis is essential for decision-making, as it indicates the minimum sales volume required to cover all operational costs.

Production and Sales Constraint

Production and sales constraints refer to the restrictions or limits imposed on the number of units that can be produced or sold. These constraints, such as a maximum capacity or sales limit, must be factored into any business analysis to ensure that the computed models are realistic and applicable to the firm's operating conditions.

Transcript

00:01 We have a problem number 90 which is related to cost function and the revenue function and the break -even point.

00:08 So this is the cost function.

00:11 This is the revenue function.

00:13 I am talking about the denominations.

00:16 And for break -even point, b -e -p break -even point, i would rather say b -e -p.

00:24 Okay.

00:26 So in this, cost function will be equal to the revenue function.

00:29 So we have been provided with a table in which fixed cost, fixed cost is $2 ,700.

00:42 Variable cost is $150 and price of item.

00:55 Price of item is $280.

01:01 So we have to find the cost function, starting with the cost function.

01:06 So cost function will always be the fixed cost that is f c let us write fc fixed cost variable cost okay fc plus vc variable cost is the cost of production per unit per unit the product okay cx will be equal to fixed cost is $2 ,700 plus 150 into x if x unit is produced so cost function will become, we should just drop the dollar sign, 2 ,700 plus 150x.

01:49 K.

01:51 Or if we take 150 as common, it will become 18, k, 18 plus x.

02:07 So this is the cost function, that it costs of producing x items.

02:15 Part b is the revenue function.

02:19 So we have cost function.

02:22 We have revenue function equal to the revenue generated from the revenue generated in producing x items...

Please give Ace some feedback