Question

1. state the number of significant digits in each of the following measurements.

(a) 5.00 cm
(b) 0.50 cm
(c) 0.500 g
(d) 0.005 g
(e) 50.0 ml
(f) 5.5 ml
(g) 50.5 °c
(h) −0.5 °c

1. perform the indicated math operation and round off the answer to the proper significant digits.

(a)
50.511 g
+ 10.25 g
(b)
97.5 g
− 95.826 g

1. perform the indicated math operation and round off the answer to the proper significant digits.

(a) (5.15 cm) (2.25 cm) (1.0 cm)
(b) $\frac{15.15\\ \text{cm}^3}{12.0\\ \text{cm}^2}$

1. explain why you round off the numbers in a calculator display after addition, subtraction, multiplication, or division of measurements.
2. (optional) a platinum cylinder has a mass of 1.000 kg, a diameter of 3.90 cm, and a height of 3.90 cm. what is the volume of the cylinder in grams per cubic centimeter? the volume of a cylinder equals $\pi d^2 h / 4$, where $\pi$ is 3.14, $d$ is the diameter, and $h$ is the height.

diagram of a cylinder with diameter $d$ and height $h$

Explanation:

Answer:

1. (a) 3

(b) 2
(c) 3
(d) 1
(e) 3
(f) 2
(g) 3
(h) 1

1. (a) 60.76 g (rounded to 60.76? Wait, 50.511 + 10.25 = 60.761, and the least decimal places is 2 (from 10.25), so round to two decimal places? Wait, no: 50.511 has three decimal places, 10.25 has two. So the sum should be rounded to two decimal places? Wait, 50.511 + 10.25 = 60.761, so rounded to two decimal places is 60.76? Wait, no, 10.25 has two decimal places, so the result should have two decimal places. So 60.76 g? Wait, no, 50.511 is 50.511 (three decimal), 10.25 is 10.25 (two decimal). When adding, the number of decimal places is determined by the least, so two decimal places. So 60.761 rounded to two decimal places is 60.76? Wait, 60.761: the third decimal is 1, so we keep the second as 6. So 60.76 g? Wait, but maybe I made a mistake. Wait, 50.511 + 10.25 = 60.761. The least number of decimal places is 2 (from 10.25), so we round to two decimal places: 60.76 g.

(b) 97.5 - 95.826 = 1.674. The least number of decimal places is 1 (from 97.5), so round to one decimal place: 1.7 g.

1. (a) (5.15)(2.25)(1.0) = 5.152.25=11.5875; 11.58751.0=11.5875. The least number of significant digits is 2 (from 1.0), so round to two significant digits? Wait, 5.15 has 3, 2.25 has 3, 1.0 has 2. So the result should have 2 significant digits? Wait, no: when multiplying, the number of significant digits is determined by the least, which is 2 (from 1.0). So 11.5875 rounded to two significant digits is 12 cm³? Wait, 11.5875 is ~12 when rounded to two significant digits (1.2 x 10¹). Wait, 11.5875: first two significant digits are 1 and 1? No, 11.5875: the first significant digit is 1, second is 1, third is 5. Wait, no, 11.5875 is 1.15875 x 10¹. So two significant digits would be 1.2 x 10¹, which is 12. So 12 cm³.

(b) 15.15 / 12.0 = 1.2625. 15.15 has 4 significant digits, 12.0 has 3. So the result should have 3 significant digits. 1.2625 rounded to three significant digits is 1.26 cm.

1. We round off to reflect the precision of the measurements. Measurements have a limited number of significant digits, indicating their precision. When performing operations, the result can’t be more precise than the least precise measurement, so we round to match the precision (number of significant digits or decimal places) of the least precise measurement.
2. First, find volume: V = πd²h/4. d = 3.90 cm, h = 3.90 cm, π = 3.14.

V = 3.14 (3.90)² 3.90 / 4
Calculate (3.90)² = 15.21
Then 3.14 * 15.21 = 47.7594
Then 47.7594 * 3.90 = 186.26166
Then divide by 4: 186.26166 / 4 = 46.565415 cm³
Mass is 1.000 kg = 1000 g
Density = mass / volume = 1000 g / 46.565415 cm³ ≈ 21.475 g/cm³. Now, check significant digits: mass is 1.000 kg (4 sig figs), d and h are 3.90 cm (3 sig figs), π is 3.14 (3 sig figs). So the least number of sig figs is 3 (from d, h, π). So density should be rounded to 3 sig figs: 21.5 g/cm³? Wait, 21.475 rounded to three significant digits is 21.5? Wait, 21.475: the third significant digit is 4, the next digit is 7, which is more than 5, so we round up the 4 to 5. So 21.5 g/cm³. Wait, but let's recalculate:
d = 3.90 (3 sig figs), h = 3.90 (3 sig figs), π = 3.14 (3 sig figs).
V = 3.14 (3.90)^2 3.90 / 4
(3.90)^2 = 15.21 (3 sig figs? Wait, 3.90 has 3, so squared is 15.21, but we keep 3 sig figs? Wait, no, when multiplying, the number of sig figs is determined by the least. So 3.90 (3) 3.90 (3) = 15.21 (we can keep it as is for intermediate steps, then round at the end). Then 3.14 (3) 15.21 (3) = 47.7594 (3 sig figs would be 47.8). Then 47.8 * 3.90 (3) = 186.42 (3 sig figs: 186). Then divide by 4: 186 / 4 = 46.5 (3 sig figs). Then mass is 1000 g (4 sig figs, but since it's 1.000 kg, which is 1000. g, so 4 sig figs). So density = 1000. g / 46.5 cm³ ≈ 21.5 g/cm³ (3 sig figs, since volume has 3). So the density is approximately 21.5 g/cm³.