Question

On average, water flows over Niagara Falls at a rate of \(2.05 \times 10^{5}\) cubic feet per second. One cubic foot of water weighs \(62.4 \mathrm{lb} .\) Calculate the rate of water flow in tons of water per day. \((1\) ton \(=2000 \mathrm{lb})\)

Short answer

The rate of water flow over Niagara Falls is \(4.524 \times 10^{9}\) tons per day.

Step-by-step solution

Convert cubic feet per second to pounds per second

First, multiply the rate of water flow in cubic feet per second by the weight of one cubic foot of water to convert the flow rate into pounds per second. The calculation is as follows: Rate in pounds per second = flow rate in cubic feet per second × weight of one cubic foot of water.

Convert pounds per second to pounds per day

Next, convert the flow rate from pounds per second to pounds per day by multiplying the rate in pounds per second by the number of seconds in a day (60 seconds/minute × 60 minutes/hour × 24 hours/day).

Convert pounds per day to tons per day

Finally, convert the daily flow rate in pounds to tons by dividing the number of pounds by the weight of one ton (2000 pounds).

Key concepts

Unit Conversion

Understanding unit conversion is essential when dealing with various measurements across diverse fields, including fluid dynamics and physical sciences. It allows us to translate quantities from one unit to another to maintain consistency, make comparisons, or perform calculations that require a specific unit.

When it comes to problems like calculating the rate of water flow, the initial step is usually to ensure that all units match for the equation to work properly. In our exercise example, converting cubic feet per second to pounds per second and eventually to tons per day involves multiplying or dividing by conversion factors that correlate the units. For instance, to convert from pounds to tons, since 1 ton equals 2,000 pounds, you simply divide the weight in pounds by 2,000.

These conversions might require dealing with large numbers, so using a calculator can help ensure accuracy. Remember, keeping track of units throughout the process is critical to arriving at the correct final answer.

Scientific Notation

Large or small numbers can be cumbersome to work with and easily lead to mistakes if not handled properly. This is where scientific notation comes in handy. It's a way to express numbers that are too big or too small to be conveniently written in decimal form. In scientific notation, numbers are written as the product of a coefficient (between 1 and 10) and a power of 10.

For example, in our exercise, the flow rate is given as 2.05 × 105 cubic feet per second. This representation keeps the number manageable and reduces the risk of calculation errors. When calculating, it's important to understand how to multiply and divide numbers in scientific notation, especially since it may involve exponents. Having a firm grasp on how to work with scientific notation is an essential skill for students tackling physics, chemistry, or any field involving large range data.

Flow Rate Measurement

Now, let's delve into the concept of flow rate measurement. Flow rate refers to the volume of fluid that passes through a given surface per unit of time. It's a critical parameter in various applications, from engineering to environmental studies.

Flow rate can be expressed in many different units, such as cubic feet per second, gallons per minute, or liters per hour. The choice of units often depends on the context and the scale of the measurement. In the case of Niagara Falls, the flow rate is exceptionally high, hence the use of cubic feet per second. To make this figure more tangible or to align it with different contexts, converting it into pounds per second or tons per day, as demonstrated in the exercise, is necessary.

Grasping flow rate measurements is not only about performing calculations correctly but also about understanding the physical meaning behind the numbers. It tells us how much water is moving and how fast which is crucial in fields such as hydrology or water resource management. By mastering these concepts, students can apply them to real-world situations and gain a deeper appreciation for the dynamics of fluid mechanics.