Question

1-6 review and reinforcement
working with numbers
complete the following statements by inserting either \are\ or \are not\ in the blanks provided.

1. zeros between two significant digits ______ significant.
2. zeros to the right of a decimal point that precede all nonzero digits ______ significant.
3. all nonzero digits ______ significant.
4. one or more final zeros used after the decimal point ______ significant.
5. zeros used solely for spacing the decimal point ______ significant.

count the number of significant digits in each of the measurements listed below. write your answer in the space provided.
significant digits

1. 230.005 m
2. 109,000 kg
3. 328.46 mm
4. 0.00607 cm³
5. 5.017 l
6. 8000 km
7. 0.057 g
8. 610.0 kpa

solve each of the following problems as directed. show all your work. watch sig figs!

1. find the volume of a cereal box that has the dimensions of 35.2 cm x 22.0 cm x 6.0 cm.
2. light travels at a speed of 2.99792458 x 10⁸ meters per second. there are 3.1557600 x 10⁷ seconds in a year. a light - year is the distance that light travels in one year. how many meters equals one light - year?

Explanation:

Step1: Recall significant - digit rules

1. Zeros between two significant digits are significant.
2. Zeros to the right of a decimal point that precede all nonzero digits are not significant.
3. All nonzero digits are significant.
4. One or more final zeros used after the decimal point are significant.
5. Zeros used solely for spacing the decimal point are not significant.

Step2: Count significant digits in each measurement

1. For 230.005 m, all digits are significant, so there are 6 significant digits.
2. For 109,000 kg, the zeros may or may not be significant. Assuming scientific - notation form, if written as \(1.09\times10^{5}\), there are 3 significant digits.
3. For 328.46 mm, all digits are significant, so there are 5 significant digits.
4. For 0.00607 \(cm^{3}\), the leading zeros are not significant, and there are 3 significant digits.
5. For 5.017 L, all digits are significant, so there are 4 significant digits.
6. For 8000 km, if written as \(8\times10^{3}\), there is 1 significant digit; if written as \(8.0\times10^{3}\), there are 2 significant digits; if written as \(8.00\times10^{3}\), there are 3 significant digits; if written as \(8.000\times10^{3}\), there are 4 significant digits. Assuming no other information, we consider 1 significant digit.
7. For 0.057 g, the leading zeros are not significant, and there are 2 significant digits.
8. For 610.0 kPa, all digits are significant, so there are 4 significant digits.

Step3: Calculate volume of cereal box

The volume \(V\) of a rectangular - box is \(V = l\times w\times h\). Given \(l = 35.2\ cm\), \(w = 22.0\ cm\), and \(h = 6.0\ cm\).
\[V=35.2\times22.0\times6.0\]
\[V = 35.2\times132\]
\[V = 4646.4\ cm^{3}\]
Rounding to the correct number of significant digits (2 for 6.0), \(V = 4600\ cm^{3}\) (in scientific notation \(4.6\times10^{3}\ cm^{3}\)).

Step4: Calculate distance of one light - year

The speed of light \(v = 2.99792458\times10^{8}\ m/s\) and the time \(t = 3.1557600\times10^{7}\ s\) in a year.
The distance \(d\) is given by \(d=v\times t\).
\[d=(2.99792458\times10^{8})\times(3.1557600\times10^{7})\]
Using the rule of exponents \(a^{m}\times a^{n}=a^{m + n}\), we have \(d=(2.99792458\times3.1557600)\times10^{8 + 7}\)
\[d = 9.46073047258\times10^{15}\ m\]
Rounding to the correct number of significant digits (8 for \(2.99792458\) and 7 for \(3.1557600\)), \(d\approx9.461\times10^{15}\ m\)

Answer:

1. are
2. are not
3. are
4. are
5. are not
6. 6
7. 3
8. 5
9. 3
10. 4
11. 1
12. 2
13. 4
14. \(4.6\times10^{3}\ cm^{3}\)
15. \(9.461\times10^{15}\ m\)