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Suppose you pay \(R\) dollars per month to rent space for the production of dolls. You pay \(c\) dollars in material and labor to make each doll, which you then sell for \(d\) dollars. a. If you produce \(n\) dolls per month, use a formula to express your net profit \(p\) per month as a function of \(R, c, d\), and \(n\). (Suggestion: First make a formula using the words rent, cost of a doll, selling price, and number of dolls. Then replace the words by appropriate letters.) b. What is your net profit per month if the rent is \(\$ 1280\) per month, it costs \(\$ 2\) to make each doll, which you sell for \(\$ 6.85\), and you produce 826 dolls per month? c. Solve the equation you got in part a for \(d\). d. Your accountant tells you that you need to make a net profit of \(\$ 4000\) per month. Your rent is \(\$ 1200\) per month, it costs \(\$ 2\) to make each doll, and your production line can make only 700 of them in a month. Under these conditions, what price do you need to charge for each doll?

Step by step solution

01

Net Profit Formula in Words

First, let's break down the net profit in terms of words. The net profit is given by: \[ \text{Net Profit} = \text{Total Revenue} - \text{Total Costs} \] where, \( \text{Total Revenue} = d \cdot n \) and \( \text{Total Costs} = R + c \cdot n \).

02

Express the Formula Using Variables

Substitute the words with the appropriate variables: \[ p = (d \cdot n) - (R + c \cdot n) \]. Simplify this to obtain: \[ p = (d - c) \cdot n - R \].

03

Net Profit Calculation

Given \( R = 1280 \), \( c = 2 \), \( d = 6.85 \), and \( n = 826 \), substitute into the formula: \[ p = (6.85 - 2) \cdot 826 - 1280 \]. Calculate: \( 4.85 \cdot 826 - 1280 \).

04

Simplify and Calculate the Profit

Calculate \( 4.85 \cdot 826 = 4008.1 \), then \( 4008.1 - 1280 = 2728.1 \). Hence, the profit is \( \$ 2728.1 \).

05

Solve for Selling Price

From Step 2, rearrange the formula to solve for \( d \): \[ p = (d - c) \cdot n - R \]. Solve for \( d \): \[ d = \frac{p + R}{n} + c \].

06

Determine Selling Price for Desired Profit

Substitute the values: \( p = 4000 \), \( R = 1200 \), \( c = 2 \), \( n = 700 \) into \( d = \frac{4000 + 1200}{700} + 2 \). Calculate: \( d = \frac{5200}{700} + 2 = 7.43 + 2 = 9.43 \). You need to charge \( \$ 9.43 \) per doll.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Functions in College Algebra

Functions are a foundational concept in college algebra. They describe a relationship between input values and output values. In simple terms, a function takes an input, processes it, and produces an output. For example, consider a function denoted by the formula \( p = (d - c) \cdot n - R \), where:

• \( d \) is the selling price per doll.
• \( c \) is the production cost per doll.
• \( n \) is the number of dolls produced.
• \( R \) is the fixed rent cost per month.

This function helps calculate the net profit by substituting the input values (known costs and prices) into the equation to find the output (profit). The relationship between all these variables and the net profit is critical in decision-making processes, especially in the context of production and sales.

Profit Calculation Simplified

Profit calculation is an essential skill for anyone dealing with business operations, and it involves determining the net income after subtracting expenses from revenue. The given exercise provides a clear example of this process. To calculate profit:

• First, identify total revenue. It is given by multiplying the selling price \(d\) by the number of items sold \(n\).
• Next, compute total costs. This includes fixed costs like rent \(R\), and variable costs like production cost \(c\) per item, multiplied by the number of items \(n\).
• Subtract the total cost from total revenue using the formula \( p = (d \cdot n) - (R + c \cdot n) \).

In the exercise, the values are provided, enabling the calculation of the profit directly. Understanding this method is valuable for managing finances or evaluating the viability of business endeavors.

Formula Manipulation Explained

Formula manipulation involves rearranging an equation to isolate a specific variable of interest. This technique is crucial for solving algebraic equations and is widely used in business calculations. In the context of the exercise, you had to solve an equation for the selling price \( d \). The formula provided is:

• Start with the net profit formula: \( p = (d - c) \cdot n - R \).
• To find \( d \), add \( R \) to both sides of the equation to get: \( p + R = (d - c) \cdot n \).
• Divide all sides by \( n \) to isolate \( (d - c) \): \( \frac{p + R}{n} = d - c \).
• Add \( c \) to both sides to solve for \( d \): \( d = \frac{p + R}{n} + c \).

This exercise demonstrates how formula manipulation can be used to solve for unknown quantities, essential for predicting outcomes like pricing strategies in business.

Analyzing Revenue and Costs

Revenue and cost analysis is vital for determining business health and strategizing future moves. In the given problem, breaking down revenue and costs helps identify how profit is generated. Here's a closer look:

• Total Revenue (TR): This is simply the product of the selling price \(d\) and the quantity sold \(n\). It represents the earnings before accounting for costs.
• Total Costs (TC): These comprise fixed costs (like rent \(R\)) and variable costs (like production cost \(c\) per item). It's crucial to account for all costs to determine true business profitability.
• Profit: Profit is obtained by subtracting total costs from total revenue. It's a measure of financial success and efficiency, calculated by \( p = (d \cdot n) - (R + c \cdot n) \).

By understanding these concepts, businesses can better manage operations and pricing strategies, ensuring sustainable profits and growth.