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Chapter 10: Problem 17

A firm uses simple linear regression to forecast the costs for its main product line. If fixed costs are equal to \(\$ 235,000\) and variable costs are \(\$ 10\) per unit, how many units does it need to sell at \(\$ 15\) per unit to make a \(\$ 300,000\) profit? a. 21,400 b. 47,000 c. 60,000 d. 107,000

Short Answer

Expert verified
The firm needs to sell 107,000 units at $\$15$ per unit to make a $\$300,000$ profit. The correct answer is (d) 107,000.

Step by step solution

01

Define the equation for Profit

The formula for Profit is given by: Profit = Sales - Costs Where Sales = Number of Units Sold * Price per Unit and Costs = Fixed Costs + (Number of Units Sold * Variable Costs). In this case, we want to achieve a profit of \(\$ 300,000\).
02

Plug in the given values

We are given the following values: - Fixed Costs = \(\$ 235,000\) - Variable Costs = \(\$ 10\) per unit - Price per Unit = \(\$ 15\) - Desired Profit = \(\$ 300,000\) We can plug those values into the profit formula: \(300,000 = (\text{Number of Units Sold} \times 15) - (235,000 + (\text{Number of Units Sold} \times 10))\)
03

Rearrange the equation to isolate Number of Units Sold

Now we need to isolate the variable 'Number of Units Sold' in the equation: \(300,000 =\; 15 \times \text{Number of Units Sold} \;-\; 235,000 \;-\; 10 \times \text{Number of Units Sold}\) Simplify and rearrange: \(300,000 + 235,000 =\; 5 \times \text{Number of Units Sold}\) \(535,000 =\; 5 \times \text{Number of Units Sold}\) Now we can divide by 5 to find the Number of Units Sold: \(\text{Number of Units Sold} =\; \frac{535,000}{5}\)
04

Solve for Number of Units Sold

Dividing 535,000 by 5, we get: \(\text{Number of Units Sold} =\; 107,000\) So, the correct answer is: d. 107,000

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Key Concepts

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

Linear Regression
Linear regression is a powerful statistical method used to model and analyze the relationships between a dependent variable and one or more independent variables. In cost accounting, it often serves the purpose of forecasting costs based on different levels of production or sales. It's a simplified way to establish how one variable affects another, creating a straight-line relationship between them.
  • Independent Variable: In the context of cost forecasting, this might be the number of units produced or sold.
  • Dependent Variable: This could be the total cost incurred by the company, which changes with the production level.
By examining past data, linear regression allows businesses to predict future expenses and optimize their budget. The primary goal is to understand how changes in the volume of goods sold affect the total cost. This helps firms make informed decisions, reducing risks related to financial planning and resource allocation.
Profit Calculation
Calculating profit is an essential function in business, crucial for assessing a firm's financial health. Profit is essentially the output after you subtract total costs from total revenue.
  • Total Revenue: The amount of money generated from selling goods or services. It's the product of the number of units sold and the price per unit.
  • Total Costs: This includes all expenses incurred, comprising fixed costs and variable costs.
The profit formula can be expressed as:\[\text{Profit} = \text{Sales} - \text{Costs}\]In the scenario provided, the goal was to find the number of units needed to attain a specific profit target. By inserting known values into the profit formula, you can rearrange to solve for the unknown variable, often the number of units sold. This systematic approach provides clarity in strategic decision-making, allowing a firm to set realistic sales targets.
Fixed and Variable Costs
Understanding the differences between fixed and variable costs is vital in cost accounting, as it forms the foundation for accurate budgeting and financial analysis.
  • Fixed Costs: These are expenses that do not change with the level of production or sales volume, such as salaries, rent, and equipment depreciation. In our example, fixed costs were $235,000 regardless of the number of units produced.
  • Variable Costs: These fluctuate with the level of production. They are directly proportional to the number of units produced and include costs like raw materials and direct labor. For our exercise, the variable cost was $10 per unit.
By dissecting these costs, businesses can better understand how scalable their operations are. Fixed costs remain constant, while variable costs will increase or decrease based on production volume. This knowledge is instrumental when setting pricing strategies and determining break-even points, impacting overall profitability.

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Most popular questions from this chapter

Perfect Fit operates a chain of 10 retail department stores. Each department store makes its own purchasing deci sions. Carl Hart, assistant to the president of Perfect Fitt, is interested in better understanding the drivers of purchasing department costs. For many years, Perrect Fit has allocated purchasing department costs to products on the basis of the dollar value of merchandise purchased. A \(\$ 100\) item is allocated 10 times as many overnead costs associated with the purchasing department as a s10 itemm. Hart recently attended a seminar titled "Cost Drivers in the Retail Industry." In a presentation at the seminar, Kaliko Fabrics, a leading competitor that has implemented activity-based costing, reported num ber of purchase orders and number of suppliers to be the two most important cost drivers of purchasing department costs. The dollar value of merchandise purchased in each purchase order was not found to be a significant cost driver. Hart interviewed several members of the purchasing department at the Perfect fitt Hart collects the following data for the most recent year for Perfect fit's 10 retail department stores: Hart decides to use simple regression analysis to examine whether one or more of three variables (the last three columns in the table) are cost drivers of purchasing department costs. Summary results for these regressions are as follows: \\[\text { Regression } 1: \mathrm{PDC}=a+(b \times \mathrm{MPS})\\] 1\. Compare and evaluate the three simple regression models estimated by Hart. Graph each one. Also, use the format employed in Exhibit \(10-18\) (page 406 ) to evaluate the information. 2\. Do the regression results support the Kaliko Fabrics' presentation about the purchasing department's cost drivers? Which of these cost drivers would you recommend in designing an ABC system? 3\. How might Hart gain additional evidence on drivers of purchasing department costs at each of Perfect Fit's stores?

Dr. Young, of Young and Associates, LLP, is examining how overhead costs behave as a function of monthly physician contact hours billed to patients. The historical data are as follows: $$\begin{array}{cc}\text { Total 0verhead costs } & \text { Physician Contact Hours Billed to Patients } \\ \hline \$ 90,000 & 150 \\\105,000 & 200 \\\111,000 & 250 \\\125,000 & 300 \\\137,000 & 350 \\\150,000 & 400\end{array}$$ 1\. Compute the linear cost function, relating total overhead costs to physician contact hours, using the representative observations of 200 and 300 hours. Plot the linear cost function. Does the constant component of the cost function represent the fixed overhead costs of Young and Associates? Why? 2\. What would be the predicted total overhead costs for (a) 150 hours and (b) 400 hours using the cost function estimated in requirement 1? Plot the predicted costs and actual costs for 150 and 400 hours. 3\. Dr. Young had a chance to do some school physicals that would have boosted physician contact hours billed to patients from 200 to 250 hours. Suppose Dr. Young, guided by the linear cost function, rejected this job because it would have brought a total increase in contribution margin of \(\$ 9,000\), before deducting the predicted increase in total overhead cost, \(\$ 10,000\). What is the total contribution margin actually forgone?

Define learning curve. Outline two models that can be used when incorporating learning into the estimation of cost functions.

A regression equation is set up, where the dependent variable is total costs and the independent variable is production. A correlation coefficient of 0.70 implies that: a. The coefficient of determination is negative. b. The level of production explains \(49 \%\) of the variation in total costs c. There is a slightly inverse relationship between production and total costs. A correlation coefficient of 1.30 would produce a regression line with better fit to the data.

Wayne Mueller, financial analyst at CELL Corporation, is examining the behavior of quarterly utility costs for budgeting purposes. Mueller collects the following data on machine-hours worked and utility costs for the past 8 quarters: $$\begin{array}{ccc} \text { Quarter } & \text { Machine-Hours } & \text { Utility costs } \\ \hline 1 & 120,000 & \$ 215,000 \\ 2 & 75,000 & 150,000 \\ 3 & 110,000 & 200,000 \\ 4 & 150,000 & 270,000 \\ 5 & 90,000 & 170,000 \\ 6 & 140,000 & 250,000 \\ 7 & 130,000 & 225,000 \\ 8 & 100,000 & 195,000 \end{array}$$ 1\. Estimate the cost function for the quarterly data using the high-low method. 2\. Plot and comment on the estimated cost function. 3\. Mueller anticipates that CELL will operate machines for 125,000 hours in quarter \(9 .\) Calculate the predicted utility costs in quarter 9 using the cost function estimated in requirement 1.
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