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. $$ \frac{d I}{d t}=100 t $$

Short answer

The capital accumulation over a five-year period is 1250 units

Step-by-step solution

Write down the integral

We are given that \(\frac{dI}{dt} = 100t\), and we know that the capital accumulation is the integral of this with respect to \(t\) over the range [0,5]. So, we write down the integral: \(\int_{0}^{5} 100t dt\)

Evaluate the integral

We now evaluate the integral. The integral of \(100t\) is \(50t^{2}\). We then evaluate this at the limits of 0 and 5. This is calculated as: \(50(5^{2}) - 50(0^{2}) = 50(25) = 1250\)

Interpret the result

Thus, the capital accumulated over a five-year period is equal to 1250 units

Key concepts

Capital Accumulation

In economics, capital accumulation is typically associated with investment in physical assets, such as machinery or real estate, that can contribute to increasing productivity or generating income. However, it can also refer to the investment in intangible assets, or simply the increase in the value of existing assets.

Capital accumulation is central to the process of economic growth and development. When businesses invest in capital, they are essentially expanding their ability to produce goods and services, which can lead to higher levels of economic output and income.

The formula \[ Capital Accumulation = \int_{0}^{5} \frac{dI}{dt} dt \] in our exercise can be interpreted as the total additional value or 'capital' that has been accumulated over a five-year period, from time \(t=0\) to \(t=5\) years, thanks to the investments made during that period. The rate of investment function \(\frac{dI}{dt} = 100t\) represents how investment changes with time, implying that investment is growing linearly with time in this scenario.

Rate of Investment

The rate of investment, denoted as \(\frac{dI}{dt}\), shows the speed at which investment is occurring with respect to time. In the context of calculus, this is often a derivative, indicating how one quantity changes as another, in this case time, changes.

In our original problem, the rate of investment was given by a simple formula: \[ \frac{dI}{dt} = 100t \] This formula tells us that the rate of investment is not constant but increases linearly with time. At the start, when \(t=0\), the rate of investment is zero, but it grows by 100 units with each passing year. The rate of investment is a crucial factor in determining the pace and volume of capital accumulation because it directs how much new capital is being generated or added to the economy over a specific period.

Integral Evaluation

Integral evaluation is a fundamental concept in calculus used to find the total accumulation of a quantity over a period. Mathematically, evaluating a definite integral means calculating the net area under the curve of a function on a given interval. In the context of our exercise, integral evaluation helps in determining the total capital accumulated over the five-year period.

The steps to evaluate the integral can be as follows: Firstly, we identify the antiderivative of the integrand \(100t\), which is \(50t^2\). This process is the reverse of differentiation. We then apply the Fundamental Theorem of Calculus, which links the definite integral with the antiderivative, to find the net area under the curve from \(t=0\) to \(t=5\), thereby calculating: \[ 50(5^2) - 50(0^2) = 1250 \] This result indicates that 1250 units of capital have been accumulated, showing the power of calculus to address practical problems in economics and finance.