The Amazon River releases 5.5 x 107 gallons of water into the Atlantic Ocean every second. There are about 3.2 x 10 seconds in a year. How many gallons are released into the ocean in one year? Show all work, and give your answer in scientific notation.
The Correct Answer and Explanation is :
To determine how many gallons of water the Amazon River releases into the Atlantic Ocean in one year, we multiply the rate of water flow per second by the number of seconds in a year.
Given:
- Water released per second: ( 5.5 \times 10^7 ) gallons
- Seconds in a year: ( 3.2 \times 10^7 ) seconds
Step 1: Multiply the two values
[
(5.5 \times 10^7) \times (3.2 \times 10^7)
]
Using the product rule of exponents:
[
a^m \times a^n = a^{m+n}
]
First, multiply the coefficients:
[
5.5 \times 3.2 = 17.6
]
Then, add the exponents:
[
10^7 \times 10^7 = 10^{14}
]
So, the result is:
[
17.6 \times 10^{14}
]
Step 2: Convert to Scientific Notation
Scientific notation requires a number between 1 and 10 multiplied by a power of 10. Since 17.6 is greater than 10, we move the decimal one place to the left, making it 1.76. To compensate, we increase the exponent by 1:
[
1.76 \times 10^{15}
]
Final Answer:
[
\mathbf{1.76 \times 10^{15} \text{ gallons per year}}
]
Explanation:
Scientific notation is a useful way to express very large or very small numbers in a compact form. In this problem, we calculated the total amount of water released into the ocean in a year by multiplying two numbers in scientific notation.
- We first multiplied the coefficients (5.5 and 3.2) to get 17.6.
- Next, we applied exponent rules to add the exponents of 10 (since the bases are the same).
- The result, ( 17.6 \times 10^{14} ), was adjusted into proper scientific notation by shifting the decimal and increasing the exponent.
This calculation demonstrates the power of scientific notation in handling real-world problems with large values efficiently.