Problem 57

Question

Why do we expect the situation to be modeled by a linear function? Give an expression for the function. The profit from making $q$ widgets is the revenue minus the cost, where the revenue is the selling price, $\$ 27$, times the number of widgets, and the cost is $\$ 1000$ for setting up a production line plus $\$ 15$ per widget.

Step-by-Step Solution

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Answer

If so, give the expression for that function. Answer: Yes, the given profit function can be modeled by a linear function. The expression for the function is P(q) = 12q - 1000.

1Step 1: Identify the profit function's components

First, we need to identify the components that make up the profit function. The profit is the revenue minus the cost. We have the following information: - Revenue: Selling price ($\$ 27$) times the number of widgets ($q$) - Cost: Setup cost ($\$ 1000$) plus per-widget cost ($\$ 15$ times the number of widgets, $q$)

2Step 2: Write the profit function

Based on the components identified in Step 1, we can write the profit function as: Profit ($P$) = Revenue - Cost $P(q) = 27q - (1000 + 15q)$

3Step 3: Simplify the profit function

To analyze the profit function and determine if it is linear, we can simplify the expression: $P(q) = 27q - 1000 - 15q$ $P(q) = (27-15) q - 1000$ $P(q) = 12q - 1000$

4Step 4: Determine if the profit function is linear

A linear function has the following form: $f(x) = mx + b$, where $m$ is the slope and $b$ is the y-intercept. Comparing the profit function, $P(q) = 12q - 1000$, to the general form of a linear function, we can see that it has the same structure: - The coefficient of $q$ ($12$) represents the slope ($m$) - The constant term ($-1000$) represents the y-intercept ($b$) Since the profit function can be expressed in the form $f(x) = mx + b$, this situation can be modeled by a linear function.

5Step 5: Give an expression for the function

The expression for the linear function representing the profit from making $q$ widgets is given by: $P(q) = 12q - 1000$

Key Concepts

Profit CalculationRevenue and Cost AnalysisSlope-Intercept Form

Profit Calculation

Calculating profit is fundamental in business operations and involves understanding revenue and costs. Profit represents the financial gain when revenue exceeds costs. In this problem, we determine the profit from selling widgets by considering both revenue and cost. Revenue is calculated by multiplying the selling price per widget ($27$ dollars) by the number of widgets ($q$). Costs include a fixed setup fee ($1000$ dollars) and a variable cost per widget ($15$ dollars times $q$). Profit ($P$) can be determined using the formula $ P(q) = \text{Revenue} - \text{Cost} $.

Identify Revenue: $27q$
Identify Costs: $1000 + 15q$
Calculate Profit: $P(q) = 27q - (1000 + 15q)$
Simplify: $P(q) = 12q - 1000$

By clearly defining these elements, profit calculation becomes straightforward. No matter the product, understanding this relationship between revenue and cost is vital for evaluating business performance.

Revenue and Cost Analysis

Analyzing revenue and cost is crucial for making informed business decisions. Let's dive deeper into each component:Revenue is the total income earned from selling goods or services. Here, it's straightforward: $27$ dollars per widget times the number of widgets, which gives us $27q$. This part shows how much money comes in from sales.Costs are what a business spends to produce goods. In this scenario, it involves two parts: the setup cost ($1000$ dollars) and the variable cost ($15$ dollars per widget). Thus, the total cost is $1000 + 15q$.

Fixed Costs: These do not change with the level of production, like $1000$ dollars for setup.
Variable Costs: Depend on the level of production, here it's $15$ dollars per widget.

Understanding how revenue and cost interact helps in setting prices, estimating profits, and deciding on production levels. It's about finding a balance to maximize profits while keeping costs manageable.

Slope-Intercept Form

The slope-intercept form is a way to describe a straight line using the equation $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. This form is very useful in understanding linear relationships.In this profit problem, the profit function $P(q) = 12q - 1000$ aligns perfectly with this form:

Slope ($m$): In $P(q) = 12q - 1000$, the slope is $12$. This tells us how much the profit changes with each additional widget sold.
Y-Intercept ($b$): The value $-1000$ represents the y-intercept, indicating the profit (or loss) when no widgets are produced or sold.

This alignment shows clearly that the profit situation can be modeled with a linear function. Recognizing the slope tells a business how quickly profits can grow with increased sales, while the y-intercept provides a baseline outlook of the financial situation if production stops.

Problem 57

Problem 58

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