Question

Using scientific notation, write the measurement \(2200 \mathrm{ft}\) as having an uncertainty of \(\pm 100 \mathrm{ft}\).

Short answer

The measurement in scientific notation with the given uncertainty is \(2.2 \times 10^3 \mathrm{ft} \pm 1 \times 10^2 \mathrm{ft}\).

Step-by-step solution

Convert the measurement to scientific notation

To convert the measurement, 2200 ft, to scientific notation, we'll rewrite it as a number between 1 and 10, multiplied by 10 raised to the power of one: \[2.2 \times 10^3 \mathrm{ft}\]

Convert the uncertainty to scientific notation

To convert the uncertainty, 100 ft, to scientific notation, we need to find a number between 1 and 10 multiplied by a power of 10 such that their product equals 100 ft. In this case, the number we are looking for is: \[1 \times 10^2 \mathrm{ft}\]

Express the entire measurement and uncertainty in scientific notation

Now we can express the measurement and its uncertainty together using scientific notation. The result is: \[2.2 \times 10^3 \mathrm{ft} \pm 1 \times 10^2 \mathrm{ft}\] This notation indicates that the measurement is 2200 ft with an uncertainty of 100 ft.

Key concepts

Measurement Uncertainty

Measurement uncertainty is a fundamental concept in science and engineering, reflecting the awareness that no measurement can be made with absolute precision. Uncertainty acknowledges the limitations of measurement tools and the variability inherent in measurement processes. It is usually expressed as a range (e.g., \( \pm 100 \mathrm{ft}\)) around a measured value to indicate the potential degree of variation from the true value.

In the exercise, the uncertainty is given as \( \pm 100 \mathrm{ft}\), meaning that the true length could be 100 feet more or less than the measured 2200 feet. By incorporating uncertainty into our measurements, we allow for a more realistic and accountable recording and analysis of data. To effectively communicate this uncertainty, incorporating it into scientific notation ensures the clarity and precision of the information being presented.

Expressing Measurements

Expressing measurements accurately is vital in communicating scientific information. It helps ensure clarity and uniform understanding among researchers, engineers, students, and the public. Often, this means using standardized units of measurement and scientific notation to represent both large and small quantities in a manageable and easy-to-read format.

For instance, in the original exercise, the measurement of 2200 feet is more compactly expressed in scientific notation as \(2.2 \times 10^3 \mathrm{ft}\). It simplifies the expression of large numbers by scaling them down to a product of a number between 1 and 10 and a power of ten. This method reduces complexities and chances for error in calculations, as well as making it easier to compare different orders of magnitude.

Powers of Ten

The concept of powers of ten is integral to scientific notation and is used to represent numbers that are much larger or smaller than can be comfortably written out in decimal form. A power of ten is essentially ten raised to a particular exponent, where an exponent indicates how many times to multiply ten by itself.

In scientific notation, a large number such as 2200 is written as \(2.2 \times 10^3\). Here, the exponent 3 means that we move the decimal place three places to the right when converting back to standard form (\(2.2\) becomes \(2200\)). Conversely, small numbers with negative exponents indicate moving the decimal to the left. By using powers of ten, information becomes more digestible, and complex calculations involving very large or small numbers become easier to perform and to understand.