Chapter 1: Problem 79
Round off or add zeros to the following calculated answers to give a final answer with three significant figures: a. \(0.00001258 \mathrm{~L}\) b. \(3.528 \times 10^{2} \mathrm{~kg}\) c. \(125111 \mathrm{~m}\) d. \(58.703 \mathrm{~g}\) e. \(3 \times 10^{-3} \mathrm{~s}\) f. \(0.010826 \mathrm{~g}\)
Short Answer
Expert verified
a. 0.0000126 L, b. 3.53 × 10² kg, c. 125000 m, d. 58.7 g, e. 3.00 × 10⁻³ s, f. 0.0108 g
Step by step solution
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
rounding significant figures
Rounding significant figures is a crucial skill in mathematics and science.
It helps ensure that the precision of your measurements and calculations is represented accurately.
When you round a number to a certain number of significant figures, you are including numbers that meaningfully contribute to its precision.
For example, let's consider 0.00001258 L.
The first three significant figures here are 1, 2, and 5.
The fourth digit, 8, tells us how to round the last significant digit.
Since 8 is greater than 5, you round 5 up by 1, making it 6.
Thus, 0.00001258 L rounded to three significant figures becomes 0.0000126 L.
Another example is 58.703 g.
Here, the first three significant figures are 5, 8, and 7.
The fourth digit is 0, which is less than 5.
Therefore, you do not need to round up, so the number remains 58.7 g when rounded to three significant figures.
It helps ensure that the precision of your measurements and calculations is represented accurately.
When you round a number to a certain number of significant figures, you are including numbers that meaningfully contribute to its precision.
For example, let's consider 0.00001258 L.
The first three significant figures here are 1, 2, and 5.
The fourth digit, 8, tells us how to round the last significant digit.
Since 8 is greater than 5, you round 5 up by 1, making it 6.
Thus, 0.00001258 L rounded to three significant figures becomes 0.0000126 L.
Another example is 58.703 g.
Here, the first three significant figures are 5, 8, and 7.
The fourth digit is 0, which is less than 5.
Therefore, you do not need to round up, so the number remains 58.7 g when rounded to three significant figures.
significant figures in measurements
In measurements, significant figures tell you how precise a measurement is.
They include all the digits you are certain about plus the first uncertain or estimated digit.
This helps convey the accuracy of your instruments and methods.
Let's take 125111 m as an example.
The first three significant figures are 1, 2, and 5.
The fourth digit is 1, which means no rounding is needed.
So you write it as 125000 m to indicate that only the first three digits are significant.
In another example, consider 0.010826 g.
The first three significant figures in this number are 1, 0, and 8.
The fourth digit is 2, so you don't need to round up.
This means the number rounded to three significant figures is 0.0108 g.
Properly noting significant figures ensures you maintain correct precision in scientific calculations.
They include all the digits you are certain about plus the first uncertain or estimated digit.
This helps convey the accuracy of your instruments and methods.
Let's take 125111 m as an example.
The first three significant figures are 1, 2, and 5.
The fourth digit is 1, which means no rounding is needed.
So you write it as 125000 m to indicate that only the first three digits are significant.
In another example, consider 0.010826 g.
The first three significant figures in this number are 1, 0, and 8.
The fourth digit is 2, so you don't need to round up.
This means the number rounded to three significant figures is 0.0108 g.
Properly noting significant figures ensures you maintain correct precision in scientific calculations.
scientific notation
Scientific notation is a way to express very large or very small numbers in a simpler form.
It is helpful in making calculations easier and more readable.
Scientific notation consists of two parts: a coefficient and an exponent (power of 10).
For instance, the number 3.528 × 10² kg can be rounded to three significant figures.
Identify the first three significant figures: 3, 5, and 2.
The fourth digit is 8, so you round 2 up by 1, making it 3.53 × 10² kg.
Similarly, a tiny number like 3 × 10⁻³ s has only one significant figure.
To write it with three significant figures, you add zeros, resulting in 3.00 × 10⁻³ s.
This makes the precision and accuracy clear, making scientific notation invaluable for handling measurements across various magnitudes.
It is helpful in making calculations easier and more readable.
Scientific notation consists of two parts: a coefficient and an exponent (power of 10).
For instance, the number 3.528 × 10² kg can be rounded to three significant figures.
Identify the first three significant figures: 3, 5, and 2.
The fourth digit is 8, so you round 2 up by 1, making it 3.53 × 10² kg.
Similarly, a tiny number like 3 × 10⁻³ s has only one significant figure.
To write it with three significant figures, you add zeros, resulting in 3.00 × 10⁻³ s.
This makes the precision and accuracy clear, making scientific notation invaluable for handling measurements across various magnitudes.
