Problem 75 - College Algebra and Calculus: An Applied Approach

Question

Present Value A company expects its income \(c\) during the next 4 years to be modeled by \(c=150,000+75,000 t\) (a) Find the actual income for the business over the 4 years. (b) Assuming an annual inflation rate of \(4 \%\), what is the present value of this income?

Step-by-Step Solution

Answer

The actual income and its present value for the company over 4 years cannot be determined without performing the integration calculations. Perform these calculations using a graphing calculator or calculus-related software to find the values.

Step 1 - Calculating the actual income

First we need to find out the actual income for the company within the span of four years. This requires the summation of income over each of the four years. This is done by calculating the definite integral of the function over the limits 0 to 4. Mathematically, this is expressed as:\[ \int_0^4 (150,000+75,000 t) \, dt \] Integrating the function will yield the total income over the 4 years.

Step 2 - Calculation

Perform the actual integration function to find the result.\[\int_0^4 (150,000+75,000 t) \, dt = 150,000t + \frac{1}{2}75,000 t^2 \, \bigg|_0^4\]Evaluate this at \(t=4\) and \(t=0\) and subtract to find the actual income over 4 years.

Step 3 - Calculating the present value

The present value of an income can be computed by discounting future inflows by an inflation rate. The formula is:\[ \int_0^4 (150,000+75,000 t)/(1+0.04)^t \, dt \]This is done by integrating this adjusted income function over the 4 year time period.

Step 4 - Calculation

You need to perform numerical integration in this case, as the exact integral is not easily computed by hand. This can be done using calculus-related software or a graphing calculator.

Key Concepts

Definite Integration

Definite integration plays a crucial role in quantifying total quantities, such as income, over a period of time. When a company wants to calculate its total income over a certain period, it takes the integral of its income function with respect to time. To conceptualize this, think of income over time as water flowing into a bucket. The rate of flow might change, but definite integration helps us measure how much water is in the bucket after a set amount of time has passed.

In the given exercise, we apply definite integration to find out the total actual income of the company over four years. The income function, expressed as c=150,000+75,000t, represents a linear increase in income, where t symbolizes the time in years. We integrate this function from 0 to 4, the span of years in consideration, which mathematically sums up the income of each infinitesimally small interval within those four years.

Future Income Modeling

Future income modeling involves using mathematical functions to predict the earnings of a business over time. This model provides a framework for understanding how a company's profits may grow and how to plan for that growth in today's value terms. In our example, the company's future income is modeled as a linear function where the income increases by a fixed amount each year.

For a deeper understanding, we see that the given function, c=150,000+75,000t, suggests that the starting income is \(150,000 and it increases by \)75,000 each year that passes. This is a simplistic model assuming a linear growth; in reality, models can be more complex and include variables such as market conditions and new revenue streams.

Inflation Rate Adjustment

Inflation diminishes the value of money over time, which means that a dollar today will not have the same purchasing power in the future. To account for this in financial calculations, we use inflation rate adjustments. When calculating present value, we adjust future income by the rate of inflation to understand what that future income is worth in today's dollars.

In the context of our exercise, assuming an annual inflation rate of 4%, we adjust the income model by discounting each year's projected income using the formula (150,000+75,000t)/(1+0.04)^t. Integrating this new expression from 0 to 4 gives us the present value of the company’s income, effectively translating future dollars into current dollars, thus providing a clearer picture of the income's current worth.