Question

Writing to Learn A dam released 1000\(\mathrm { m } ^ { 3 }\) of water at 10\(\mathrm { m } ^ { 3 / \mathrm { min } }\) and then released another 1000\(\mathrm { m } ^ { 3 }\) at 20\(\mathrm { m } ^ { 3 / \mathrm { min } }\) . What was the average rate at which the water was released? Give reasons for your answer.

Short answer

The average rate at which the water was released is approximately 13.33\(\mathrm { m } ^ { 3 / \mathrm { min } }\). The calculations were done by finding the time it took to release the water at each rate, summing these times to find the total time and then dividing the total volume of water released by this total time.

Step-by-step solution

Find the Time It Took to Release Water at Each Rate

The time taken to release water at a certain rate can be calculated by dividing the volume of water released by the rate of release. Therefore, the time taken for the first 1000\(\mathrm { m } ^ { 3 }\) was \( \frac { 1000 }{10} = 100 \mathrm { min } \), and for the second 1000\(\mathrm { m } ^ { 3 }\) was \( \frac { 1000 }{20} = 50 \mathrm { min } \).

Calculate the Total Time Taken

The total time for the release of all the water is given by the sum of the times calculated in step 1. Therefore, the total time is \( 100 \mathrm { min } + 50 \mathrm { min } = 150 \mathrm { min } \).

Calculate the Average Rate

The average rate at which the water was released can be calculated by dividing the total volume of water released by the total time taken. Therefore, the average rate is \( \frac { 2000\mathrm { m } ^ { 3 } }{150\mathrm { min } } = 13.33\mathrm { m } ^ { 3 / \mathrm { min } }\).

Key concepts

Calculus in Real-World Problems

Calculus, the mathematical study of continuous change, is an essential tool in understanding and describing the dynamic world around us. In real-world applications, calculus helps us analyze situations where quantities change at varying rates over time.

For instance, the exercise about a dam releasing water involves calculating the rate of change of the water's volume with respect to time. This type of problem is directly related to the concept of derivatives in calculus, which measures how a quantity changes as another quantity, typically time, changes.

Understanding the calculus behind such scenarios enables us to predict how long it will take for a certain quantity of water to be released at varying rates, which can be instrumental in planning and resource management in environmental and engineering contexts.

Rate of Release Calculation

Calculating the rate of release in scenarios like a dam releasing water is quite common, not only in textbooks but also in practical engineering problems. The rate at which a substance is released, distributed, or consumed is essential for numerous industries and applications.

To determine the rate properly, one needs to understand the mathematical relationship between volume and time. In our example, the rate of water flow is measured in cubic meters per minute. By calculating the duration for each release and summing up these intervals, we obtain the total time the process takes.

Then, by dividing the total volume released by the total time it took for the release, we can establish the average rate of release. This skill is invaluable when it comes to managing resources efficiently or predicting how changes in rates of release will impact the duration of a process.

Volume and Rate Relationship

The relationship between volume and rate is fundamental in fluid dynamics and many other fields of science and engineering. In our dam release example, it's evident how rates can affect volumes over time.

The exercise improvement advice focuses on understanding that volume divided by rate gives us the time each phase of release takes. This may seem intuitive, but it's important to grasp that if the rate of release is doubled, the time it takes to release a certain volume of water is halved.

This inverse relationship indicates a crucial concept in calculus and physics: the notion of proportional change and how adjusting one variable impacts another. By dissecting the problem in terms of volume and rate, students can better predict outcomes and understand the implications of rate adjustments in real-world scenarios.