Problem 61 - Algebra 1

Question

In Exercises \(58-60\), use the following information. In \(1803,\) the Louisiana Purchase added \(8.28 \times 10^{5}\) square miles to the United States. The cost of this land was \(\$ 15\) million. In \(1853,\) the Gadsden Purchase added \(2.94 \times 10^{4}\) square miles, and the cost was \(\$ 10\) million. WATERFALL Stanley Falls in Congo, Africa, has an average flow of about \(1.7 \times 10^{4}\) cubic meters per second. How much water goes over Stanley Falls in a typical 30-day month?

Step-by-Step Solution

Answer

In a typical 30-day month, about \(4.4064 \times 10^{10}\) cubic meters of water go over Stanley Falls.

Step 1: Understand the Problem

In this problem, we need to find out how much water goes over Stanley Falls in Congo, Africa, in a typical 30-day month. We are given that the average flow of water is \(1.7 \times 10^{4}\) cubic meters per second.

Step 2: Convert Time Horizons

First, figure out how many seconds are in a 30-day month. A day has 24 hours, each hour has 60 minutes, and each minute has 60 seconds, so a day has \(60 \times 60 \times 24 = 86,400\) seconds. Therefore, a 30-day month has \(30 \times 86,400 = 2,592,000\) seconds.

Step 3: Calculate Water Volume

Multiply the number of seconds in a month by the flow rate per second to find the total volume of water. That is \(1.7 \times 10^{4} \text{ cubic meters/second} \times 2,592,000 \text{ seconds} = 4.4064 \times 10^{10}\) cubic meters.

Key Concepts

Exponential Notation

Exponential notation, also known as scientific notation, is a mathematical expression used to simplify numbers that are either very large or very small. It represents a number as a product of a decimal number between 1 and 10, multiplied by a power of 10. This form is convenient for expressing the enormous range of values we encounter in science and mathematics, such as the flow rate of water over Stanley Falls mentioned in the exercise.

The general form of exponential notation is: \( a \times 10^{n} \), where \( a \) is the decimal number (the coefficient) and \( n \) is the exponent (the power of 10). In the example of Stanley Falls' flow rate, \(1.7 \times 10^{4}\) cubic meters per second indicates that the 1.7 should be multiplied by 10 raised to the power of 4, which moves the decimal point 4 places to the right, resulting in 17,000 cubic meters per second.

Unit Conversion

Unit conversion is a fundamental principle in many areas of mathematics and science and it is essential for solving problems like the one pertaining to Stanley Falls' water volume. The process involves multiplying by unit fractions that equal one, allowing us to switch from one unit to another without changing the value of the measurement.

In the given problem, we have to convert time from days to seconds, which requires understanding that 1 day equals 24 hours, 1 hour equals 60 minutes, and 1 minute equals 60 seconds. Here's a breakdown:

• 1 day = 24 hours
• 1 hour = 60 minutes
• 1 minute = 60 seconds

Through this stepwise conversion, the seconds in a month are calculated, and the exercise further illustrates how to use this in the more extensive problem of determining the total volume of water passing over Stanley Falls.

Rate Problems in Algebra

Rate problems in algebra involve finding how quantities relate to each other over time, and they often require a combination of unit conversion and exponential notation to solve. In the Stanley Falls problem, we're dealing with a rate of water flow (cubic meters per second) over a given time (30 days). We typically resolve these problems by identifying the rate, establishing the time and then multiplying the two to find the total quantity.

To solve the original exercise, one must:

• Identify the rate of flow: \(1.7 \times 10^{4}\) cubic meters per second.
• Convert the time period into an equivalent unit that matches the rate unit (seconds).
• Multiply the rate by the total number of seconds in a 30-day month to find the volume of water.

The result is a quantifiable answer measuring the amount of water that flows over Stanley Falls in a typical month, highlighting both the application of algebra in real-world scenarios and the importance of understanding rates and unit conversion.