A useful and easy-to-remember approximate value for the number of seconds in a year is π × 107. Determine the percent error in this approximate value. (There are 365.24 days in one year.)

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First, calculate the exact number of seconds in a year. Start by determining the number of seconds in a day: there are 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Multiply these together: 24 × 60 × 60.

Next, calculate the total number of seconds in a year using the exact number of days in a year, which is 365.24 days. Multiply the number of seconds in a day by the number of days in a year: 365.24 × (24 × 60 × 60).

Now, calculate the approximate number of seconds in a year using the given approximation: π × 10^7.

Determine the percent error by comparing the approximate value to the exact value. Use the formula for percent error: ((approximate value - exact value) / exact value) × 100%.

Finally, substitute the values you calculated into the percent error formula and simplify to find the percent error.

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주요 개념

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Percent Error

Percent error is a measure of how inaccurate a value is compared to a true or accepted value, expressed as a percentage. It is calculated using the formula: Percent Error = |(Approximate Value - True Value) / True Value| × 100%. This concept helps quantify the deviation of an estimated value from the actual value.

Seconds in a Year

To determine the true number of seconds in a year, multiply the number of days in a year (365.24) by the number of hours in a day (24), minutes in an hour (60), and seconds in a minute (60). This calculation provides the precise number of seconds, which is essential for comparing against the approximate value given in the problem.

Approximation Using π

The approximation π × 10^7 is a mnemonic for estimating the number of seconds in a year. π is approximately 3.14159, and when multiplied by 10 million, it provides a rough estimate. Understanding this approximation helps in evaluating its accuracy and calculating the percent error compared to the precise calculation.