Arizona State University| PHY 121: Univ Physics I: Mechanics|Homework 1-1| Using Significant Figures

Learning Goal: To practice using significant figures

You can think of a significant figure as being a digit that is reliably known. For example, a length measurement of 6.2 cm has two significant figures, the 6 and the 2. The next decimal place–the one-hundredths–is not reliably known and is thus not a significant figure.

Determining the proper number of significant figures is straightforward, but there a few definite rules to follow. These are summarized in the following:

  1. When multiplying or dividing several numbers, or when taking square roots, the number of significant figures in the answer should match the number of significant figures of the least precisely known number in the calculation. 
  2. When adding or subtracting several numbers, the number of decimal places in the answer should match the smallest number of decimal places of any number used in the calculation.
  3. Exact numbers are perfectly known and do not affect the number of significant figures an answer should have. Examples of exact numbers are the 2 and the π in the formula C=2πr for the circumference of a circle.
  4. It is acceptable to keep one or two extra digits during intermediate steps of a calculation, to minimize rounding error, as long as the final answer is reported with the proper number of significant figures.

Problem

PART A. Your bedroom has a rectangular shape, and you want to measure its area. You use a tape that is precise to 0.001 m and find that the shortest wall in the room is 3.547 m long. The tape, however, is too short to measure the length of the second wall, so you use a second tape, which is longer but only precise to 0.01 m. You measure the second wall to be 4.79 m long. Which of the following numbers is the most precise estimate that you can obtain from your measurements for the area of your bedroom?

A. 17.0 m²

B. 16.990 m²

C. 16.99 m²

D. 16.9 m²

E. 16.8 m²

ANSWER: 17.0 m²

When you calculate the area of your bedroom, you have to multiply the measurements of the lengths of the walls. Considering that the measurements have different precision, the answer should match the number of significant figures of the least precisely known number in the calculation which is 3 significant digits.

PART B. Using the measurements described in Part A, which of the following numbers is the most precise estimate for the perimeter of your bedroom?

A. 16.674 m

B. 16.67 m

C. 16.68 m

D. 16.7 m

E. 17 m

ANSWER: 16.67 m

When you calculate the perimeter of your bedroom, you have to add the measurements of the lengths of all the walls. Considering that the measurements have different precision, he answer should match the smallest number of decimal places of any number used in the calculation which is 2 decimal places

PART C. If your bedroom has a circular shape, and its diameter measured 6.32 m, which of the following numbers would be the most precise value for its area?

A. 30 m²

B. 31.0 m²

C. 31.4 m²

D. 31.37 m²

E. 31.371 m²

ANSWER: 31.4 m²

To calculate the area A of a circle given its diameter d, you need to first divide the diameter by 2, then multiply your result by itself (or take the square of it), and finally multiply everything by π:

A=\pi \left(\frac{d}{2}\right)^2

Since here the number 2 and π are exact numbers, they do not change the accuracy of the measured numbers involved in the calculation. Therefore, your answer should be expressed to the same number of significant figures as that used in the given diameter.

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