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Chapter 3: Problem 3

A firm estimates that its sales will rise by \(3 \%\) each year and that it needs to sell at least 10000 goods each year in order to make a profit. Given that its current annual sales are only 9000 , how many years will it take before the firm breaks even?

Short Answer

Expert verified
It will take approximately 4 years.

Step by step solution

01

- Identify the Growth Formula

The firm's sales are growing at a constant percentage rate each year. This situation can be modeled using the compound growth formula: \[ S = P (1 + r)^t \] where \( S \) is the future sales, \( P \) is the current sales, \( r \) is the annual growth rate, and \( t \) is the number of years.
02

- Substitute Given Values

We are given that the current sales \( P \) are 9000, the growth rate \( r \) is 3% or 0.03, and we want to find the time \( t \) when the sales \( S \) will be at least 10000. Substituting these into the formula, we get: \[ 10000 = 9000 (1 + 0.03)^t \]
03

- Solve for \( t \)

To solve for \( t \), we first divide both sides of the equation by 9000: \[ \frac{10000}{9000} = (1 + 0.03)^t \] which simplifies to: \[ \frac{10}{9} = (1.03)^t \] Next, we'll take the natural logarithm of both sides to isolate \( t \): \[ \ln \left( \frac{10}{9} \right) = \ln \left( (1.03)^t \right) \] Using the property of logarithms that \( \ln(a^b) = b \ln(a) \), we get: \[ \ln \left( \frac{10}{9} \right) = t \ln(1.03) \]
04

- Calculate \( t \)

To find \( t \), we divide both sides by \( \ln(1.03) \): \[ t = \frac{\ln \left( \frac{10}{9} \right)}{\ln(1.03)} \] Using a calculator to compute the values: \[ \ln \left( \frac{10}{9} \right) \approx 0.1054 \] \[ \ln(1.03) \approx 0.0296 \] Therefore: \[ t = \frac{0.1054}{0.0296} \approx 3.56 \]
05

- Interpret the Result

Since \( t \) represents the number of years and must be a whole number, we round up to the nearest whole number. Hence, it will take about 4 years for the firm's sales to reach at least 10000 units and break even.

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Key Concepts

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

Growth Rate
The growth rate is a percentage that indicates how much something increases over time. In our context, the firm's sales grow at a rate of 3% each year. You can think of it like a compound interest rate where each year the sales increase by 3% of the previous year's sales.
A higher growth rate means faster growth of sales. The growth rate can be represented using the variable \( r\), and it must be converted into decimal form for calculations (3% becomes 0.03).
Current Sales
Current sales represent the amount of goods sold by the firm in the present year. Our problem states that the firm currently sells 9000 units annually. This number can be denoted as \( P \) in the compound growth formula.
Understanding your current sales is essential because it serves as the baseline for future growth projections. Without knowing where you start (current sales), it鈥檚 impossible to figure out future growth (future sales).
Natural Logarithm
The natural logarithm (ln) is a mathematical function often used in growth and decay problems. It's the inverse of the exponential function, meaning \( \text{if} \ e^x = y \text{ then} \ln(y) = x \).
In our solution, we use the natural logarithm to solve for the number of years (\( t\)) needed for the firm's sales to grow from 9000 to 10000 units.
Here鈥檚 why natural logarithms are useful:
  • They simplify equations that involve exponentiation.
  • They help us isolate the variable we need to solve for.
To take the natural logarithm of both sides, we use the property \( \ln(a^b) = b \ln(a) \), allowing us to solve for \( t \).
Break-Even Analysis
Break-even analysis helps determine when a firm will start making a profit. In this problem, the break-even point is when the firm's annual sales reach 10000 units, allowing it to cover all costs and begin profiting.
The key information we needed to conduct this analysis were:
  • Current annual sales (\( P = 9000 \))
  • Annual growth rate (\( r = 0.03 \))
  • Required sales to break even (\( S = 10000 \))
Using the compound growth formula \[S = P (1 + r)^t \], we substitute the given values and solve for \( t \) - this gives us the number of years required to reach the break-even point.
Because time is always whole-number years, we round up, finding it will take about 4 years for the firm's sales to break even and start making a profit.

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Most popular questions from this chapter

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