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Chapter 2: Problem 8

Wind Up Corporation manufactures widgets. The monthly expense equation is \(E=3.20 q+56,000\) . They plan to sell the widgets to retailers at a wholesale price of \(\$ 6.00\) each. How many widgets must be sold to reach the break even point?

Short Answer

Expert verified
Wind Up Corporation needs to sell 20,000 widgets to reach the break-even point.

Step by step solution

01

Define Total Revenue and Total Expense

The Revenue (R) equation is derived from the selling price of the widget (\$6.00) multiplied by the quantity of widgets (q). Thus, the equation is \(R = 6q\).The Expense (E) equation is given as \(E = 3.20q + 56000\). Now, we need to set these two equations equal to each other to solve for the break-even point.
02

Formulate the Equilibrium Equation and Solve for q

For the break-even point, Revenue equals Expense, hence \(6q = 3.20q + 56000\).We solve for q by first subtracting \(3.20q\) from both sides of the equation, which gives us \(2.80q = 56000\). Then, we divide both sides by 2.80 to isolate q, yielding \(q = 56000 / 2.80\).
03

Calculate the Break-Even Quantity of Widgets

By plugging the numbers into our calculator, we find that \(q = 20000\). This means that the Wind Up Corporation must sell 20,000 widgets to reach the break-even point.

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Key Concepts

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

Expense Equation
When a company creates a product, various expenses are associated with its production. In this example, Wind Up Corporation has an expense equation given by \(E=3.20q+56,000\). This equation is essential for determining the cost of producing a specified number of widgets (denoted by \(q\), which stands for quantity). Here, there are two components to the expense equation:
  • Variable Costs: \(3.20q\), which means for each widget produced, it costs \(\$3.20\).
  • Fixed Costs: \(56,000\), which represents costs that remain constant regardless of how many widgets are produced.
Understanding this equation helps companies predict and plan for their financial outlays.
Revenue Equation
The revenue equation captures the income generated from selling goods or services. In this case, each widget is sold at \(\\(6.00\). Consequently, the revenue equation for Wind Up Corporation is \(R = 6q\). This implies:
  • **For each widget sold, the company earns \(\\)6.00\).
  • **The total revenue is directly proportional to the number of widgets sold.
This equation is crucial as it allows the company to forecast its total potential earnings.
Algebraic Equations
Solving algebraic equations often encompasses finding a specific value, or range of values, that make the equation true. In the break-even analysis, we set the expense equation equal to the revenue equation: \(6q = 3.20q + 56000\). Here are the steps to solve it:
  • We subtract \(3.20q\) from both sides, simplifying it to \(2.80q = 56000\).
  • Next, divide both sides by \(2.80\) to isolate \(q\).
  • This results in \(q = 20000\) widgets.
These steps illustrate the application of basic algebra to solve real-world financial problems.
Financial Algebra
Financial algebra blends traditional algebraic concepts with financial real-world applications. The break-even analysis is a perfect example. It combines the cost of creating a product (expense equation) with the revenue from selling that product (revenue equation) to determine when a business neither profits nor loses money.
  • The break-even point provides businesses with critical insights into the minimum number of units they need to sell to cover their costs.
  • It aids in financial planning and setting sales goals.
Understanding financial algebra helps businesses make informed decisions about pricing, production, and sales optimization.

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Most popular questions from this chapter

A bicycle sells for a retail price of b dollars from an online store. The wholesale price of the bicycle is w. a. Express the markup algebraically. b. Express the percent increase of the markup algebraically.

A supplier of school kits has determined that the combined fixed and variable expenses to market and sell G kits is W. a. What expression models the price of a school kit at the breakeven point? b. Suppose a new marketing manager joined the company and determined that the combined fixed and variable expenses would only be 80% of the cost if the supplier sold twice as many kits. Write an expression for the price of a kit at the breakeven point using the new marketing manager鈥檚 business model.

Flip Flops manufactures beach sandals. Their expense and revenue functions are \(E=-300 p+32,000\) and \(R=-275 p^{2}+6,500 p\) . a. Determine the profit function. b. Determine the price, to the nearest cent, that yields the maximum profi t. c. Determine the maximum profit, to the nearest cent.

Green yard鈥檚 manufactures and sells yard furniture made out of recycled materials. It is considering making a lawn chair from recycled aluminum and fabric products. The expense and revenue functions are \(E=-1,850 p+800,000\) and \(R=-100 p^{2}+20,000 p.\) a. Determine the profit function. b. Determine the price, to the nearest cent, that yields the maximum profit. c. Determine the maximum profit, to the nearest cent.

Wanda's Widgets used market surveys and linear regression to develop a demand function based on the wholesale price. The demand function is \(q=-140 p+9,000\) . The expense function is \(E=2.00 q+16,000\) .a. Express the expense function in terms of \(p\) . b. At a price of \(\$ 10.00\) , how many widgets are demanded? c. How much does it cost to produce the number of widgets from part b?
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