Question

Carry out the following calculations, making sure that your answer has the correct number of significant figures: (a) \(\frac{2.420 \mathrm{~g}+15.6 \mathrm{~g}}{4.8 \mathrm{~g}}\) (b) \(\frac{7.87 \mathrm{~mL}}{16.1 \mathrm{~mL}-8.44 \mathrm{~mL}}\) (c) \(V=\pi r^{2} h,\) where \(r=6.23 \mathrm{~cm}\) and \(h=4.630 \mathrm{~cm}\)

Short answer

a) 3.8 , b) 1.03 , c) 565 cm^3.

Step-by-step solution

Addition in part (a)

Add the masses: \(2.420 \text{ g} + 15.6 \text{ g} = 18.02 \text{ g}\)This result has 4 significant figures since the least number of significant figures in the numbers being added is 4 (for 15.6 g).

Division in part (a)

Divide the combined mass by the given mass: \(\frac{18.02 \text{ g}}{4.8 \text{ g}} = 3.753333...\)The result should be rounded to 2 significant figures because the mass 4.8 g has the least number of significant figures.Thus, the final answer is 3.8.

Subtraction in part (b)

Subtract the volumes: \(16.1 \text{ mL} - 8.44 \text{ mL} = 7.66 \text{ mL}\)The result has 3 significant figures, same as the number subtracted (8.44 mL).

Division in part (b)

Divide the given volume by the result of the subtraction: \(\frac{7.87 \text{ mL}}{7.66 \text{ mL}} = 1.02739...\)The result should be rounded to 3 significant figures because the result of the subtraction (7.66 mL) has the least number of significant figures.Thus, the final answer is 1.03.

Calculations for part (c)

Calculate the volume using the formula \(V = \pi r^2 h\):\(\begin{align*} r &= 6.23 \text{ cm} \ h &= 4.630 \text{ cm} \ V &= \pi (6.23 \text{ cm})^2 (4.630 \text{ cm}) \ &= \pi (38.8129 \text{ cm}^2)(4.630 \text{ cm}) \ &= \pi (179.770067 \text{ cm}^3) \ &= 564.9903382 \text{ cm}^3 \ \end{align*}\)This needs to be rounded to 3 significant figures because the radius has the least number of significant figures (6.23 cm).Thus, the final answer is \(565 \text{ cm}^3\).

Key concepts

Addition and Subtraction Rules

When doing addition or subtraction with significant figures, it's important to focus on the decimal places rather than the total number of significant figures. For example, in the exercise, we add 2.420 g and 15.6 g together. Here, the number with the least decimal places (15.6 g) determines the number of decimal places in the result. This results in a sum of 18.02 g. The final answer must have the same number of decimal places (one in this case) as the number with the least decimal places.

Division Rules

When dividing numbers, the result must have the same number of significant figures as the number with the fewest significant figures in the original calculation. For instance, in part (a) of the problem, we divide 18.02 g by 4.8 g. The number 4.8 has only 2 significant figures, so the final answer should also be rounded to 2 significant figures. Hence, the division yields 3.8.

Volume Calculation

To calculate the volume, especially involving shapes like cylinders, you typically use the formula: \( V = \pi r^2 h \). Here, \( r \) is the radius, and \( h \) is the height. Given \( r = 6.23 \) cm and \( h = 4.630 \) cm in the exercise, you would first calculate the squared radius \( (6.23)^2 = 38.8129 \) cm\textsuperscript{2}. Multiplying this by the height gives 179.770067 cm\textsuperscript{3}, and multiplying with \( \pi \) approximates to 564.9903382 cm\textsuperscript{3}. The radius had the fewest significant figures (three in this case), so the final rounded volume is 565 cm\textsuperscript{3}.

Significant Figures in Multiplication and Division

When performing multiplication and division, it's crucial to round your result to have the same number of significant figures as the number with the least significant figures in your original numbers. For example, in part (b) of the exercise, we divide 7.87 mL by 7.66 mL. The number 7.66 mL has 3 significant figures, so the division result must also be rounded to 3 significant figures, yielding 1.03.