- Multiply the weight per gallon by the total number of gallons to find the initial product: \(8.3 \times 6.6 \times 10^5 = 54.78 \times 10^5\).
- Convert the result into standard scientific notation by adjusting the coefficient so it lies between 1 and 10.
- Shift the decimal point one place to the left and increase the exponent of 10 by one: \(54.78 \times 10^5 = 5.478 \times 10^6\).
- The final weight of the water is \(\boxed{5.478 \times 10^6\text{ lb}}\).
Explanation
- Analyzing the problem and identifying data
The objective is to find the total weight of water in an Olympic-sized swimming pool, given the weight per gallon and the volume of the pool. We are provided with the following data:
- Weight per gallon of water: \(8.3\text{ lb}\).
- Volume of the pool: \(6.6 \times 10^5\text{ gallons}\).
To find the total weight, we need to multiply the weight per unit volume by the total volume of the water in the pool.
- Calculating the total weight
The total weight \(W\) is calculated by the product of the weight per gallon and the total number of gallons:
\[W = 8.3\text{ lb/gallon} \times (6.6 \times 10^5\text{ gallons})\]
First, we multiply the numeric coefficients:
\[8.3 \times 6.6 = 54.78\]
Substituting this back into the expression, we get:
\[W = 54.78 \times 10^5\text{ lb}\]
- Converting to standard scientific notation and concluding the result
Standard scientific notation requires the coefficient to be a number between 1 and 10. Currently, our coefficient is 54.78, which is greater than 10. We rewrite it as:
\[54.78 = 5.478 \times 10^1\]
Now, substitute this into our expression for total weight:
\[W = (5.478 \times 10^1) \times 10^5 = 5.478 \times 10^{1+5} = 5.478 \times 10^6\text{ lb}\]
This result matches the third option provided in the problem.
Examples
Understanding scientific notation is essential in fields like physics, chemistry, and engineering where we work with very large or very small quantities. For example, astronomers use scientific notation to express the massive distances between stars or the weight of planets, while chemists use it to represent the number of atoms in a tiny sample (Avogadro's number). This helps to simplify calculations and makes large numbers easier to compare and communicate.