Problem 62 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Maxwell started a home theater business in 2005 . The revenue of his company for that year was $$\$ 240,000$$. The revenue grew by \(20 \%\) in 2006 and by \(30 \%\) in 2007 . Maxwell projected that the revenue growth for his company in the next 3 yr will be at least \(25 \% /\) year. How much does Maxwell expect his minimum revenue to be for \(2010 ?\)

Step-by-Step Solution

Answer

The minimum revenue for 2010 is \(Revenue_2007 * (1 + 0.25)^3\), which is approximately $775{,}569.60.

Step 1: Find the revenue for 2006

To find the revenue for 2006, we need to calculate 20% of the initial revenue in 2005 and add it to the 2005 revenue. Revenue_2006 = Initial revenue (2005) + 20% of Initial revenue Revenue_2006 = \(240{,}000 + 0.20 \times 240{,}000\) Now, calculate the revenue for 2006.

Step 2: Find the revenue for 2007

To find the revenue for 2007, we need to calculate 30% of the revenue we found in the previous step (2006's revenue) and add it to the 2006 revenue. Revenue_2007 = Revenue_2006 + 30% of Revenue_2006. Revenue_2007 = Revenue_2006 (1 + 0.30) Now, calculate the revenue for 2007.

Step 3: Calculate the compounded growth

We are given a minimum growth rate of 25% per year for the next 3 years. We can use compound interest formula to calculate the compounded growth A = P(1 + r)^n Where: - A = the future value of the investment/loan, including interest - P = the principal investment amount (the initial deposit or loan amount) - r = the annual interest rate (decimal) - n = the number of times that interest is compounded per unit t In our case, - r = 25% = 0.25 - n = 3 (3 years) Now we calculate the compounded growth.

Step 4: Find the minimum revenue for 2010

To find the minimum revenue for 2010, we apply the compounded growth that we just calculated to the revenue in 2007. Minimum_revenue_2010 = Revenue_2007 * A Now calculate the minimum revenue for 2010.

Key Concepts

Revenue Growth

Revenue growth is the increase in revenue over a specific period. In Maxwell's case, starting with a revenue of $240,000 in 2005, we see an increase over the following years. Understanding revenue growth is crucial for businesses as it reflects the overall health and expansion of the company.
Here's how to think about revenue growth:

• Year-by-Year Analysis: Look at each year's revenue independently, comparing it against the previous year to understand the growth trend.
• Percentage Increase: Calculate the percentage increase each year by finding how much the revenue has increased from the previous year and dividing by the previous year’s revenue.
• Long-Term Projection: Consider multi-year projections to plan for future growth and understand long-term business viability.

For Maxwell, revenue growth was 20% in 2006 and 30% in 2007, indicating a positive trend. For the years ahead, projecting a consistent growth of 25% indicates a robust forecast. By understanding these calculations, Maxwell can plan for business expansion and allocate resources appropriately.

Compound Interest

Compound interest is a powerful concept in finance that allows gains to be calculated on both the initial principal and the accumulated interest from previous periods. This exponential growth can greatly benefit businesses looking to grow their revenues.
The formula for compound interest is given by:\[ A = P(1 + r)^n \]where:

• \(A\) is the amount of money accumulated after n years, including interest
• \(P\) is the principal amount (initial revenue)
• \(r\) is the annual interest rate (expressed as a decimal)
• \(n\) is the number of years

In Maxwell's scenario, he projects a minimum revenue growth rate of 25% per year for three years. Using the compound interest formula, you find how this growth impacts his business. By substituting the necessary values, this formula calculates the compounded future revenue, providing insight into his financial progression.

Percent Increase

Percent increase is a mathematical concept used to measure how much a quantity grows compared to its original value. It is an essential tool for businesses aiming to understand and communicate their growth effectively.
The formula to find the percentage increase is:

• Calculate the increase: \( \text{Increase} = \text{New Value} - \text{Original Value} \)
• Find the percentage: \( \frac{\text{Increase}}{\text{Original Value}} \times 100 \)

In practice, for the year 2006, Maxwell's business grew from $240,000 by 20%. Calculating: \( \text{Increase} = 20\% \times 240{,}000 \), then adding the increase to the original revenue gives the new value.
In subsequent years, this method applies similarly to assess year-over-year performance. With knowledge of the percent increase, businesses can track and strategize better around sustaining and boosting growth.