Problem 105 - Algebra A Combined Function

Question

Although the actual amount varies by season and time of day, the average volume of water that flows over Niagara Falls (the American and Canadian falls combined) each second is \(7.5 \times 10^{5}\) gallons. How much water flows over Niagara Falls in an hour? Write the result in scientific notation. (Hint: 1 hour equals 3600 seconds.) (Source: niagarafallslive.com)

Step-by-Step Solution

Answer

The volume is \(2.7 \times 10^{8}\) gallons in an hour.

Step 1: Understand the Problem

We need to calculate the total volume of water flowing over Niagara Falls in an hour. We know the flow rate of water is \(7.5 \times 10^{5}\) gallons per second.

Step 2: Determine Total Time

Convert the time from hours to seconds because the flow rate is given in seconds. There are 3600 seconds in an hour.

Step 3: Calculate Total Water Flow

Multiply the flow rate per second by the total number of seconds in an hour: \(7.5 \times 10^{5}\, \text{gallons/second} \times 3600\, \text{seconds}\).

Step 4: Simplify the Multiplication

First multiply the coefficients: \(7.5 \times 3600 = 27000\). Convert this to scientific notation to get \(2.7 \times 10^{4}\).

Step 5: Adjust the Powers of Ten

Since the original multiplication was \(7.5 \times 10^{5}\) and \(3600\) can be considered as \(3.6 \times 10^{3}\), add the exponents of 10: \(10^{5} \times 10^{3} = 10^{8}\).

Step 6: Combine Results in Scientific Notation

Now, combine \(2.7\) from the multiplication and \(10^8\) from the powers of ten to get \(2.7 \times 10^{8}\).

Step 7: Conclusion

The volume of water that flows over Niagara Falls in an hour is \(2.7 \times 10^{8}\) gallons.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are particularly useful when dealing with very large or small numbers.
In the context of scientific notation, exponential functions help simplify and represent these numbers succinctly.

• **Base:** The number that is repeatedly multiplied. For scientific notation, this is usually 10.
• **Exponent:** The power to which the base is raised. It shows how many times the base is used in the multiplication.
• **Expression Example:** In the expression \(7.5 \times 10^{5}\), 10 is the base and 5 is the exponent.

Understanding these concepts is vital as they allow us to manipulate numbers more easily.
In practical applications, by adding exponents, like \(10^{5} \) and \(10^{3} \) becoming \(10^{8} \), we can perform calculations step by step to manage large values effectively.

Unit Conversion

Unit conversion involves changing values from one unit of measurement to another.
This concept is crucial when dealing with problems that involve rates, such as water flow per unit of time, especially when the units of measurement don't initially match.

• **Time Conversion:** Through the hint given in the problem, we know that 1 hour equals 3600 seconds.
• **Expression in Seconds:** If a given rate is per second, as in the exercise, breaking down time into seconds ensures consistency.

When performing unit conversion, always ensure:

• **Correct Units:** Check that all parts of the equation are using the same units for accurate results.
• **Step-by-Step Calculation:** Avoid errors by controlling each step of the conversion thoroughly.

This process is essential for ensuring that subsequent calculations remain accurate and understandable.

Multiplication of Large Numbers

Multiplying large numbers can be challenging due to their size and complexity. Thankfully, scientific notation offers a straightforward method to tackle these calculations with ease.
Here's how you can approach multiplying large numbers like in the exercise:

• **Separate Multiplication:** First, multiply the coefficients (the numbers in front). For instance, \(7.5 \times 3600 = 27000\).
• **Scientific Notation Simplification:** Convert the result into scientific notation to simplify it further. Here, \(27000 = 2.7 \times 10^{4}\).
• **Combine Exponents:** Multiply the powers of 10 separately. When combining, add the exponents, such as \(10^{5} \times 10^{3} = 10^{8}\).
• **Final Combination:** Coupling the simplified coefficient and the combined power of ten gives the overall result as \(2.7 \times 10^{8}\).

This method allows you to manage cumbersome numbers without losing accuracy, ensuring you arrive at the solution systematically and efficiently.