The water inflow rate into a storage tank is not constant but increases over time. Initially, the pump delivers water at a rate of 5 L/min. However, the inflow rate increases by 2 L/min for each additional minute due to rising pressure or system adjustments. This scenario can be described mathematically by a linear function:
It is necessary to integrate the inflow rate function to measure the total volume of water added to the tank over time. The total water volume V(t) is obtained by performing the following integral:
When the power rule is applied to evaluate the integral, both quadratic and linear terms emerge, reflecting the changing rate of inflow:
The term t2 arises due to the increasing inflow rate, demonstrating that water accumulation follows a quadratic pattern over time. The term 5t represents the steady inflow component that exists from the beginning. The constant C accounts for the initial volume of water in the tank before the inflow process begins. Since integration determines the total volume by summing the instantaneous rate of change, this equation provides a comprehensive function that describes the total amount of water in the tank at any given moment.
Water flows into a large industrial storage tank, but the inflow rate is not constant. Initially, the pump delivers water at 5 m3/s.
The water inflow is increasing at a constant rate of two cubic meters per second due to continuous system adjustments.
This steady change makes the water inflow rate a linear function of time, where 2 is the slope showing the rate of increase, and 5 is the initial inflow rate.
The goal is to find the total volume of water stored in the tank at any point in time. This requires indefinite integration, which finds the total volume of water by integrating the rate of water inflow over time.
Instead of analyzing inflow at specific instants, integration provides a single function for the total volume of water with respect to time.
Integrating the linear inflow rate introduces a quadratic term, which shows the accelerating growth of the water volume, while the linear term shows the constant growth from the initial inflow rate.
Finally, the constant of integration is given by the volume function at t equals zero, which corresponds to the initial volume of water present in the tank.
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