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Chapter 1: Problem 80

Carry out the following calculation, and report the answer to the correct number of significant figures. $$ (1.68)\left[\frac{23.56-2.3}{1.248 \times 10^{3}}\right] $$

Short Answer

Expert verified
The answer is 0.0287, rounded to three significant figures.

Step by step solution

01

Perform the Subtraction in the Numerator

In the expression \(23.56 - 2.3\), subtract the numbers while keeping in mind significant figures rules. The number 2.3 has two significant figures, so the result should also have two significant figures. Calculation: \(23.56 - 2.3 = 21.26\) However, since we must adhere to the least number of decimal places, which is one decimal place from 2.3, the result should be \(21.3\).
02

Divide the Result by the Denominator

Now take the result from the numerator and divide by the denominator: \(\frac{21.3}{1.248 \times 10^3}\)Perform the division: \(21.3 \div 1248 = 0.0170673076923077\)The denominator (1.248) limits the significant figures here, which has four significant figures, so the answer should be rounded to four significant figures: \(0.01707\)
03

Multiply by the Number Outside the Brackets

Now, multiply the bracket result by 1.68. Calculation: \(1.68 \times 0.01707 = 0.0286776\)The number 1.68 has three significant figures, so round the result to three significant figures: \(0.0287\).
04

Present the Final Result with Correct Significant Figures

The final computation after multiplication should be presented with the correct significant figures as determined in the previous steps:\[0.0287\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subtraction with Significant Figures
When performing subtraction, significant figures play a crucial role. The guiding rule for subtraction is that the result should have the same number of decimal places as the measurement with the fewest decimal places. This helps in maintaining the precision of the least precise value in the calculation.

In this exercise, we subtracted 2.3 from 23.56. Even though the computation gives us 21.26, we must round to reflect the number with the least decimal places, which is one from 2.3. Therefore, the result is correctly reported as 21.3. An easy way to remember this rule is: "Stick to the lowest decimal places, just like glue!"
  • Always check the number of decimal places, not the number of significant figures in subtraction.
  • Round your result to have the same decimal precision as the least precise measurement.
Properly handling significant figures in subtraction ensures accuracy and consistency across calculations.
Division with Scientific Notation
When dividing numbers, especially those involving scientific notation, significant figures again become essential. The number of significant figures in the result is determined by the original number with the least significant figures.

In the provided solution, after subtracting, the quotient \(\frac{21.3}{1.248 \times 10^3}\) provides a valuable insight. The denominator (1.248) has four significant figures. Thus, despite the detailed decimal answer being 0.0170673, you should only keep four significant figures, rounding it to 0.01707.
  • Count the significant figures in each element involved in the division.
  • Match the result's significant figures to the component with the fewest significant figures.
  • Using scientific notation correctly can help in making calculations more manageable and less error-prone.
Scientific notation not only helps in dealing with very large or small numbers but also keeps significant figures in check in a busy calculation.
Multiplication with Significant Figures
In multiplication, similar to division, the significant figures in the result should match those of the input number with the least significant figures. This ensures that the precision of your answer is aligned with the measured data.

In the final step of the exercise, we multiply 1.68 by 0.01707. Here, 1.68, having three significant figures, determines that our answer must be rounded to also reflect three significant figures. Thus, the raw result of 0.0286776 is appropriately rounded to 0.0287.
  • Observe the significant figures in all numbers being multiplied.
  • Limit the significant figures in your final result to match the smallest significant figures in your inputs.
  • Always perform rounding only at the very end of your calculation to avoid cumulative errors.
By applying these principles, one can consistently deliver accurate results where precision mirrors the least precise measurement used.

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Most popular questions from this chapter

A red blood cell has a diameter of \(7.5 \mu \mathrm{m}\) (micrometers). What is this dimension in (a) meters, (b) nanometers, and (c) picometers?

A compact disk has a diameter of \(11.8 \mathrm{cm} .\) What is the surface area of the disk in square centimeters? In square meters? [Area of a circle \(\left.=(\pi) \text { (radius) }^{2} .\right]\)

To determine the average mass of a popcorn kernel you collect the following data: $$\begin{array}{ll} \hline \text { Number of kernels } & \text { Mass }(\mathrm{g}) \\ \hline 5 & 0.836 \\ 12 & 2.162 \\ 35 & 5.801 \\ \hline \end{array}$$ Plot the data with number of kernels on the \(x\) -axis and mass on the y-axis. Draw the best straight line using the points on the graph (or do a least- squares or linear regression analysis using a computer program) and then write the equation for the resulting straight line. What is the slope of the line? What does the slope of the line signify about the mass of a popcorn kernel? What is the mass of 50 popcorn kernels? How many kernels are there in a handful of popcorn \((20.88 \mathrm{g}) ?\)

The fluoridation of city water supplies has been practiced in the United States for several decades. It is done by continuously adding sodium fluoride to water as it comes from a reservoir. Assume you live in a medium-sized city of 150,000 people and that \(660 \mathrm{L}(170 \mathrm{gal})\) of water is consumed per person per day. What mass of sodium fluoride (in kilograms) must be added to the water supply each year (365 days) to have the required fluoride concentration of 1 ppm (part per million)-that is, 1 kilogram of fluoride per 1 million kilograms of water? (Sodium fluoride is \(45.0 \%\) fluoride, and water has a density of \(1.00 \mathrm{g} / \mathrm{cm}^{3} .\) )

Some soft drinks are sold in bottles with a volume of \(1.5 \mathrm{L}\) What is this volume in milliliters? In cubic centimeters? In cubic decimeters?
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