91Ó°ÊÓ

Inflow and outflow rates are crucial in understanding how materials enter and leave a system. In this exercise, we have a tank where water enters through two pipes and exits via one pipe. To determine the net effect, we need to calculate the total inflow rate and subtract the outflow rate.

The inflow rates of the two inlet pipes are given as 0.6 \text{kg/s} and 0.7 \text{kg/s}. Adding these together, we get: \(0.6 \text{kg/s} + 0.7 \text{kg/s} = 1.3 \text{kg/s}\).

This is the total rate at which water enters the tank. The outflow rate is given as 1 \text{kg/s}. To find the net inflow rate, subtract the outflow rate from the total inflow rate:

\(1.3 \text{kg/s} - 1 \text{kg/s} = 0.3 \text{kg/s}\).

This net inflow rate tells us that the tank gains water at a rate of 0.3 kg per second.

Mass balance is the principle of conservation of mass, which states that mass cannot be created or destroyed. We apply this principle to track how the mass of water in the tank changes over time. Starting with 500 kg of water in the tank, we account for the difference between the inflow and outflow rates to determine the net balance of water.

In this problem, the net inflow rate is 0.3 kg/s. Over twenty minutes, the mass change in the tank can be found by multiplying this rate by the time duration in seconds:

\(20 \text{minutes} \times 60 \text{seconds/minute} = 1200 \text{seconds}\).

Now, multiply the net inflow rate by this duration:

\(0.3 \text{kg/s} \times 1200 \text{seconds} = 360 \text{kg}\).

This tells us that 360 kg of water is added to the tank during this period. Adding this to the initial amount of water in the tank, we get the final mass:

\(500 \text{kg} + 360 \text{kg} = 860 \text{kg}\).

Thus, after twenty minutes, the tank will contain 860 kg of water.

Unit conversion is essential in solving problems where different units are used for time, mass, or any other measurement. In this exercise, we need to convert time from minutes to seconds to use consistent units throughout our calculations.

The problem specifies a duration of twenty minutes. Since most rates are given per second, converting minutes to seconds helps align the units for all calculations:

\(20 \text{minutes} \times 60 \text{seconds/minute} = 1200 \text{seconds}\).

This conversion ensures we can accurately determine how much water enters and exits the tank over the specified period. Consistent units allow straightforward multiplication of the net inflow rate by the total time in seconds, yielding the mass change directly in kilograms. Understanding and applying unit conversions correctly is key to achieving accurate results in such problems.