Question

1.12 • the volume of a solid cylinder is given by v = πr²h, where r is the radius and h is the height. you measure the radius and height of a thin cylindrical wire and obtain the results r = 0.036 cm and h = 12.1 cm. what do your measurements give for the volume of the wire in mm³? use the correct number of significant figures in your answer. 1.13 •• a useful and easy - to - remember approximate value for the number of seconds in a year is π×10⁷. determine the percent error in this approximate value. (there are 365.24 days in one year.)

Explanation:

Step1: Convert units for 1.12

First convert radius and height to mm. $r = 0.036\ cm=0.36\ mm$, $h = 12.1\ cm = 121\ mm$.

Step2: Calculate volume for 1.12

Use the volume formula $V=\pi r^{2}h$. Substitute $r = 0.36\ mm$ and $h = 121\ mm$ into it.
$V=\pi\times(0.36)^{2}\times121$
$V=\pi\times0.1296\times121$
$V\approx49.4\ mm^{3}$ (using 3 significant - figures as 0.36 has 2 significant figures and 121 has 3 significant figures, we follow the least number of significant figures in multiplication)

Step3: Calculate actual number of seconds in a year for 1.13

There are 365.24 days in a year, 24 hours in a day, 60 minutes in an hour and 60 seconds in a minute.
The actual number of seconds in a year $N_{actual}=365.24\times24\times60\times60$
$N_{actual}=365.24\times86400$
$N_{actual}=31556900$ seconds

Step4: Calculate approximate number of seconds in a year for 1.13

The approximate number of seconds in a year $N_{approx}=\pi\times10^{7}\approx3.14\times10^{7}=31400000$ seconds

Step5: Calculate percent - error for 1.13

The percent - error formula is $\text{Percent Error}=\frac{\vert N_{actual}-N_{approx}\vert}{N_{actual}}\times100\%$
$\text{Percent Error}=\frac{\vert31556900 - 31400000\vert}{31556900}\times100\%$
$\text{Percent Error}=\frac{156900}{31556900}\times100\%$
$\text{Percent Error}\approx0.50\%$

Answer:

1.12: $49.4\ mm^{3}$
1.13: $0.50\%$