Problem 82 - Algebra A Combined Function

Question

An environmental technician finds that warm water from an industrial process is being discharged into a nearby pond at a rate of 30 gallons per minute. Plant regulations state that the flow rate should be no more than 0.1 cubic feet per second. Is the flow rate of 30 gallons per minute in violation of the plant regulations? (Hint: 1 cubic foot is equivalent to 7.48 gallons.)

Step-by-Step Solution

Answer

The flow rate of 30 gallons per minute does not violate plant regulations.

Step 1: Understand the Given Flow Rate

We are given that the discharge rate is 30 gallons per minute. We need to convert this into a unit that's compatible with the regulation, which is in cubic feet per second.

Step 2: Conversion Factor

The conversion factor between gallons and cubic feet is provided: 1 cubic foot is equivalent to 7.48 gallons. Hence, to convert from gallons to cubic feet, divide the number of gallons by 7.48.

Step 3: Convert Gallons per Minute to Cubic Feet per Minute

To find the flow rate in cubic feet per minute, use the conversion factor. It is calculated as follows:\[\text{Flow rate in cubic feet per minute} = \frac{30 \text{ gallons per minute}}{7.48} \approx 4.01 \text{ cubic feet per minute}\]

Step 4: Convert Cubic Feet per Minute to Cubic Feet per Second

Since regulations specify the rate in cubic feet per second (cfs), you need to convert cubic feet per minute to cubic feet per second by dividing by 60 (since there are 60 seconds in a minute):\[\text{Flow rate in cubic feet per second} = \frac{4.01 \text{ cubic feet per minute}}{60} \approx 0.0668 \text{ cfs}\]

Step 5: Compare with Regulation Limit

The calculated flow rate \(0.0668 \text{ cfs}\) is compared with the maximum allowed flow rate set by the plant regulation which is 0.1 cubic feet per second. Since \(0.0668 < 0.1\), the flow rate complies with the regulations.

Key Concepts

Flow Rate

When dealing with liquids in industrial processes, one of the critical factors to consider is the flow rate. Flow rate refers to the volume of fluid that passes through a given surface or area over a specific period of time. It's often used to assess how much fluid is being released or moved within a system. In our everyday activities, understanding flow rates is crucial, whether for plumbing systems, irrigation, or industrial applications. Here are some key considerations for flow rate:

• Measured in volume per unit time, such as gallons per minute (gpm) or cubic feet per second (cfs).
• Helps ensure systems operate within safety and efficiency guidelines.
• Can be used to determine if processes are adhering to environmental regulations.

Determining the appropriate flow rate involves not just measuring the fluid's speed but also its volume. For businesses and industries, adhering to prescribed flow rates can mean staying compliant with environmental standards, as in the case of the industrial discharge into the pond.

Cubic Feet per Second

Cubic feet per second (cfs) is a unit of measurement often applied to quantify flow rates. It is particularly useful in contexts involving large volumes of water, such as rivers or wastewater discharges. When regulations specify discharge rates, cubic feet per second is a common standard due to its clear indication of how much space is being filled by water each second. Understanding cubic feet per second:

• One cubic foot is the volume within a cube of 1 foot on each side. When combined with the unit of seconds, it provides both volume and time factors.
• Allows for easy comparison of flow rates to see if they are above or below specific thresholds.
• Used widely in environmental engineering to monitor and control water flow for safety and compliance.

In our exercise scenario, measuring the industrial discharge in cfs provides a precise indication of whether or not the flow rate is in line with regulations. Converting from other units, like gallons per minute, to cfs can help in making accurate assessments.

Gallons to Cubic Feet Conversion

The conversion between gallons and cubic feet is crucial in many engineering and environmental disciplines. Understanding how to convert units correctly can mean the difference between compliance and violation of rules, as liquid measurements often need to shift from one system of units to another depending on the context. Here's how the conversion works:

• The given conversion factor is that 1 cubic foot equals 7.48 gallons. This is a standard measurement that simplifies the process of converting volumes.
• To convert from gallons to cubic feet, you divide the number of gallons by 7.48.
• Similarly, to convert from cubic feet to gallons, you multiply the number of cubic feet by 7.48.

In our example, converting the discharge from gallons per minute to cubic feet per second follows these steps. Starting with gallons per minute, convert it into cubic feet per minute using the 7.48 factor, and further adjust for seconds to meet regulation standards. This ensures all parties have a common understanding and can assess compliance accurately.