Problem 11 - Applied Calculus

Question

A village wishes to measure the quantity of water that is piped to a factory during a typical morning. A gauge on the water line gives the flow rate (in cubic meters per hour) at any instant. The flow rate is about \(100 \mathrm{~m}^{3} / \mathrm{hr}\) at \(6 \mathrm{am}\) and increases steadily to about \(280 \mathrm{~m}^{3} / \mathrm{hr}\) at \(9 \mathrm{am} .\) Using only this information, give your best estimate of the total volume of water used by the factory between 6 \(\mathrm{am}\) and \(9 \mathrm{am}\).

Step-by-Step Solution

Answer

The estimated total volume of water used is 570 cubic meters.

Step 1: Understand the Flow Rate Increase

We have a flow rate that changes from 100 cubic meters per hour at 6 am to 280 cubic meters per hour at 9 am. This change is linear because it increases steadily. Therefore, we can assume the flow rate changes at a constant rate over time.

Step 2: Determine Linear Increase

To find the increase per hour, calculate the difference in flow rates between the start and end times. The increase is \(280 - 100 = 180 \; \text{m}^3/\text{hr}\). This occurs over 3 hours, so the rate of increase per hour is \( \frac{180}{3} = 60 \; \text{m}^3/\text{hr}\).

Step 3: Calculate Flow Rates for Each Hour

The flow rate increases by 60 cubic meters per hour. At 7 am, the flow rate is \(100 + 60 = 160 \; \text{m}^3/\text{hr}\), and at 8 am, it is \(160 + 60 = 220 \; \text{m}^3/\text{hr}\). At 9 am, the flow rate is \(220 + 60 = 280 \; \text{m}^3/\text{hr}\).

Step 4: Approximate using Average Flow Rate

Using the estimated flow rates, calculate the average flow rate for the 3-hour period. The average flow rate is \(\frac{100 + 280}{2} = 190 \; \text{m}^3/\text{hr}\).

Step 5: Calculate Total Water Volume

To find the total volume of water used, multiply the average flow rate by the total time period of 3 hours: \[ 190 \; \text{m}^3/\text{hr} \times 3 \; \text{hr} = 570 \; \text{m}^3 \].

Key Concepts

Linear Increase

In the context of flow rate, a linear increase means that the flow rate changes at a constant pace over time. This is an essential concept as it allows us to use simple arithmetic to predict flow rates for future times.
The problem involves determining the water flow from 6 am to 9 am, where the flow rate steadily increases from 100 to 280 cubic meters per hour. Since it's linear, the increase is uniform, meaning the same amount is added to the flow rate every hour.

• We start at 100 cubic meters per hour at 6 am.
• By 9 am, we reach 280 cubic meters per hour.
• The entire increase over these 3 hours is 180 cubic meters per hour (280 - 100).

This steady rate of change can easily be divided by the time interval, allowing us to find out how much the flow rate changes each hour, specifically 60 cubic meters per hour. Recognizing this consistent change simplifies our calculations for flow estimation.

Average Flow Rate

To simplify the calculation of total volume, it is often useful to determine the average flow rate during a given period. The average flow rate acts as a constant placeholder to approximate a variable rate over time.
For any given hour, we take the starting and ending flow rates and average them. In this case, the average flow rate over three hours is calculated by:

• The flow starts at 100 and ends at 280 cubic meters per hour.
• Average flow rate = (start + end) / 2 = (100 + 280) / 2 = 190 cubic meters per hour.

This calculation suggests that if water flowed at 190 cubic meters per hour consistently over 3 hours, the total volume would be the same as the one with an increasing flow rate. Averaging offers a simpler way to estimate the overall impact when dealing with linear or variable rates.

Volume Estimation

Understanding volume estimation is crucial in calculating the total usage of water when dealing with varying flow rates. By determining the correct calculations, we can accurately estimate the total water volume utilized over a specific period.
The formula used is: **Volume** = Average Flow Rate x Time Interval.

• With an average flow rate of 190 cubic meters/hour,
• and a time span of 3 hours, the total volume can be calculated as: \[ 190 imes 3 = 570 \; \text{m}^3 \]

This methodology provides a straightforward path to evaluate total usage by leveraging both time and average rate. It underscores the importance of accurate rate assessment for effective volume calculation, particularly in situations with linearly increasing flow rates.