Problem 89 - College Algebra and Calculus: An Applied Approach

Question

You are given the rate of investment \(d l / d t\). Find the capital accumulation over a five-year period by evaluating the definite integral Capital accumulation \(=\int_{0}^{5} \frac{d l}{d t} d t\) where \(t\) is the time in years. $$ \frac{d I}{d t}=500 $$

Step-by-Step Solution

Answer

The capital accumulation over a five-year period is 2500 units.

Step 1: Identify the given parameters

Given rate of investment, \( \frac{dI}{dt} = 500 \), time period from \( t = 0 \) to \( t = 5 \) years.

Step 2: Formulate the integral to solve

The capital accumulation over a five-year period is given by the definite integral \( \int_{0}^{5} \frac{dI}{dt} dt \).

Step 3: Evaluate the integral

Using basic rules of integration, ∫dt = t. So, the integral ∫500dt from 0 to 5 is \( 500 \int_{0}^{5} dt = 500[t]_{0}^{5} \) which simplifies to \( 500(5 - 0) \).

Step 4: Calculate the result

Now perform the multiplication: 500 * 5 = 2500.

Key Concepts

Rate of Investment

The rate of investment is a fundamental concept in finance that describes how quickly investments are made over time. This rate is often denoted by the derivative of investment over time, which in mathematical terms is expressed as \( \frac{dI}{dt} \).
In our exercise, the rate of investment \( \frac{dI}{dt} \) is given as a constant value, 500. This implies a steady inflow of investments at this constant rate over the time period considered. To understand this better:

• The rate of investment tells you how much investment occurs per unit of time, in this case, each year.
• A constant rate, such as 500, simplifies calculations as the rate doesn't change over the years considered.

When dealing with problems involving rate of investment, the goal is often to determine how these investments accumulate over a specified time period.

Capital Accumulation

Capital accumulation refers to the growth of capital resources over time. It measures the total value of investments that have been amassed. In this problem, we're tasked with determining the amount of capital that accumulates over a five-year period.
Using the rate of investment, we calculate this by evaluating a definite integral.To break this down:

• You're given a time period from \( t = 0 \) to \( t = 5 \).
• The integral \( \int_{0}^{5} \frac{dI}{dt} dt \) calculates the total increase in capital based on the continuous rate of investment.

The outcome of this integral, which we calculated to be 2500, indicates the total amount of capital that accumulates, given the constant investment rate. This means, over these five years, your investments total 2500 units of currency or resources.

In practical terms, capital accumulation is crucial as it reflects the financial growth and potential of investments over time.

Integration

Integration is a key mathematical process to solve problems involving continuous rates and accumulations. It involves calculating the total area under a curve described by a function. In this exercise, integration helps us translate the rate of investment, a derivative, into capital accumulation, a total amount.Here's how it works:

• The integral symbol \( \int \) denotes the process of integration, providing a method to find accumulated quantities over time or space.
• For constant functions, such as our \( \frac{dI}{dt} = 500 \), integration becomes relatively straightforward as it boils down to multiplying the rate by the interval's length.

Mathematically, integrating the rate function \( \int_0^5 500 \, dt \) yields \( 500[t]_0^5 = 500(5 - 0) = 2500 \). This simple operation highlights how integration helps convert a continuous rate into tangible quantities.

In economics and finance, integration helps understand how small continuous changes, like investment rates, accumulate over time, offering insights into long-term financial planning.