Purpase: To practice expressing significant figures. (b) A rectangular block of wood is measured to have the dimensions 11.2 cm×3.4 cm× 4.10 cm. Compute the volume of the block, showing explicitly (by underlining) how doubtful figures’are carried through the calculation, and report the final answer with the correct number of significant figures. Calculations Computed volume (show work) (in powers of 10 notation) (units) (c) In an experiment to determine the value of p, a cylinder is measured to have an average value of 4.25 cm for its diameter and an average value of 13.39 cm for its circumference. What is the experimental value of p to the correct number of significant figures? Colculations (show work) Experimental value of p (units)
The Correct Answer and Explanation is:
(b) Volume of a Rectangular Block
Given dimensions:
- Length = 11.2 cm (3 significant figures)
- Width = 3.4 cm (2 significant figures)
- Height = 4.10 cm (3 significant figures)
Volume = Length × Width × Height V=11.2 cm×3.4 cm×4.10 cmV = 11.2\ \text{cm} \times 3.4\ \text{cm} \times 4.10\ \text{cm} V=(11.2×3.4)=38.08‾ cm2V = (11.2 \times 3.4) = \underline{38.08}\ \text{cm}^2 V=38.08‾×4.10=156.128‾ cm3V = \underline{38.08} \times 4.10 = \underline{156.128}\ \text{cm}^3
The least number of significant figures in the given measurements is 2 (from 3.4 cm), so the final answer must have 2 significant figures.
Final Answer: V=1.6×102 cm3V = 1.6 \times 10^2\ \text{cm}^3
(c) Experimental Value of π
Given:
- Diameter = 4.25 cm (3 significant figures)
- Circumference = 13.39 cm (4 significant figures)
We use the formula: π=CircumferenceDiameter=13.39 cm4.25 cm=3.150588‾…\pi = \frac{\text{Circumference}}{\text{Diameter}} = \frac{13.39\ \text{cm}}{4.25\ \text{cm}} = \underline{3.150588}\ldots
The diameter has the least number of significant figures (3), so the final answer should be rounded to 3 significant figures.
Final Answer: π=3.15 (unitless)\pi = 3.15\ (\text{unitless})
Explanation
When performing scientific calculations, it’s essential to understand and apply the rules of significant figures (sig figs) to express results with the appropriate precision. Significant figures reflect the precision of measured values and help avoid overrepresenting the certainty of a calculated result.
In part (b), we are given the dimensions of a rectangular block and asked to compute its volume. The multiplication involves values with different levels of precision. Specifically, the width (3.4 cm) has only 2 significant figures, which is the limiting factor. During multiplication or division, the number of significant figures in the final answer must match the smallest number of significant figures in any of the values used. While the intermediate steps are calculated with full precision (e.g., 11.2 × 3.4 = 38.08), we underline the doubtful figures to track the precision. Finally, we round 156.128 cm³ to 2 significant figures, giving 1.6 × 10² cm³.
In part (c), we calculate an experimental value for π by dividing the measured circumference by the diameter. The circumference has 4 significant figures, while the diameter has 3. Hence, the result must be reported to 3 significant figures. The unrounded division gives 3.150588…, which is then rounded to 3.15, reflecting the precision of the input data.
Maintaining correct significant figures ensures that your results are not misleadingly precise. It demonstrates a solid understanding of the limitations of measurement tools and experimental uncertainty.
