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Chapter 1: Problem 65

Assume the linear cost and revenue models applies. An item costs \(\$ 3\) to make. If fixed costs are \(\$ 1000\) and profits are \(\$ 7000\) when 1000 items are made and sold, find the revenue equation.

Short Answer

Expert verified
The revenue equation is \(R(x) = 11x\).

Step by step solution

01

Understand the Problem

We are given that the cost to make each item is \(3 and the fixed costs are \)1000. The profit when selling 1000 items is $7000. We need to find the revenue equation, which is typically of the form \(R(x) = mx\), where \(x\) is the number of items.
02

Calculate Total Costs

First, calculate the total cost \(C(x)\) for producing \(x\) items. Since each item costs \(3 to make and fixed costs are \)1000, the cost equation is:\[C(x) = 3x + 1000\]
03

Determine Profit Equation

Profit is the difference between revenue and cost. The profit equation is given by \(P(x) = R(x) - C(x)\). We know that when 1000 items are sold, the profit is $7000, giving us:\[7000 = R(1000) - C(1000)\]
04

Find Revenue for 1000 Items

Substitute the cost equation into the profit equation for 1000 items:\[C(1000) = 3(1000) + 1000 = 4000\]\[7000 = R(1000) - 4000\]Therefore, \[R(1000) = 7000 + 4000 = 11000\]
05

Write Revenue Equation

Since we found \(R(1000) = 11000\) and revenue is typically given by a linear equation \(R(x) = mx\), we can conclude that the revenue per item (\(m\)) is \(\frac{11000}{1000} = 11\). So the revenue equation is:\[R(x) = 11x\]

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Key Concepts

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

Fixed Costs
Fixed costs are the expenses that do not change with the number of items produced or sold. They remain constant across various levels of production. In this exercise, the fixed costs are given as $1000.
  • Fixed costs include things like rent, salaries, and other overhead expenses that are incurred regardless of production levels.
  • They are crucial because they must be paid even if no items are produced.
Understanding fixed costs is essential for businesses to know how much they need to earn from sales before they start making a profit. In our exercise, these fixed costs form part of the total cost calculation, seen in the equation \[C(x) = 3x + 1000\]. This equation shows that aside from the variable cost \(3x\), we always have an additional 1000 dollars to account for.
Profit Calculation
Profit is the positive financial gain after all expenses and costs are subtracted from the total revenue. In the given exercise, profit is found by subtracting the total cost from the revenue generated by selling the items.
  • The profit equation is given by \( P(x) = R(x) - C(x) \), where \(P(x)\) is profit, \(R(x)\) is revenue, and \(C(x)\) is cost.
  • For 1000 items, the problem states that the profit is $7000. This gives us the equation \(7000 = R(1000) - C(1000)\).
  • From the solution, we know that \(C(1000) = 4000\), resulting in \[7000 = R(1000) - 4000\]. Solving this gives \( R(1000) = 11000 \).
Profit calculation helps businesses determine how efficiently they are operating. The aim of any business is to maximize this profit by either increasing revenue or decreasing costs.
Revenue Equation
The revenue equation represents the total income generated from business activities. In a linear revenue model, the revenue is proportional to the number of items sold. In this problem, we develop a revenue equation based on a specific number of items sold.
  • A linear revenue equation is generally expressed as \( R(x) = mx \), where \(m\) represents the revenue generated per item and \(x\) is the number of items.
  • After determining from the exercise solution that \( R(1000) = 11000 \), the revenue per item is calculated as \( \frac{11000}{1000} = 11 \).
  • Thus, the revenue equation becomes \( R(x) = 11x \).
This equation helps businesses predict income levels for different production scales. A clear understanding of the revenue per item is essential for setting sales targets and pricing products effectively.

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Most popular questions from this chapter

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