Chapter 1, Problem 65QP

Carry out the following conversions: (a) 185 nm to meters, (b) 4.5 billion years (roughly the age of Earth) to seconds (assume 365 days in a year), (c) $\text{71}{\text{.2 cm}}^{\text{3}}$ to cubic meters, (d) $\text{88}{\text{.6 m}}^{\text{3}}$ to liters.

Interpretation Introduction

Interpretation:

The given conversions are to be carried out for the given measurements.

Concept introduction:

Conversion factor is a fraction obtained from a relationship between the units. It is written as a ratio, and can be inverted to give two conversion factors for every relationship.

$1\text{m =}{\text{10}}^{-9}\text{nm}$, to convert nm to m, the conversion factor is $\frac{\text{1 m}}{{\text{10}}^9\cancel{\text{nm}}}$.

$1\text{ hr}=60\text{ min}$, to convert hours to minutes, the conversion factor is $\frac{60\text{ min}}{1\text{ hr}}$.

$1\text{ min}=60\text{ s}$, to convert minute to second, the conversion factor is $\frac{60\text{s}}{1\min }$.

$\text{1 mile}=\text{1609 m}$, to convert meter to miles, the conversion factor is $\frac{\text{1 mile}}{\text{1609 m}}$.

$\text{1 m = 100 cm}$, to convert centimeter to meter, the conversion factor is $\frac{\text{1 m}}{\text{100 cm}}$.

${\text{1 m}}^3\text{ = 1000 L}$, to convert cubic meter to liters, the conversion factor is $\frac{\text{1000 L}}{{\text{1 m}}^3}$.

Answer to Problem 65QP

Solution:

a) $\text{1}\text{.85}\times {\text{10}}^{-7}\text{m}$

b) $\text{1}{\text{.4×10}}^{17}\text{s}$

c) $\text{7}\text{.12}\times {\text{10}}^{-5}{\text{ m}}^3$

d) $\text{8}\text{.86}\times {\text{10}}^4\text{ L}$

Explanation of Solution

a) 185 nm to meter

Convert the quantity $\text{185 nm}$ to meters.

Set up the unit conversion calculation using the appropriate conversion factor, which is as follows:

$\left(\text{185 }\cancel{\text{nm}}\left(\frac{\text{1 m}}{{\text{10}}^9\cancel{\text{nm}}}\right)\right)=\text{1}\text{.85}\times {\text{10}}^{-7}\text{m}$.

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{185 nm}=\text{1}\text{.85}\times {\text{10}}^{-7}\text{m}$.

b) $4.5$ billion years to seconds

Convert the quantity $4.5$ billion years to seconds.

Set up the unit conversion calculation using the appropriate conversion factors.

The conversion is as follows:

$\left(4.5\times {10}^9\cancel{\text{years }}\text{× }\frac{365\bcancel{\text{days}}}{1\cancel{\text{year}}}\text{ × }\frac{24\text{ h}}{1\bcancel{\text{day}}}\times \frac{60\cancel{\text{min}}}{1\text{ h}}\text{×}\frac{\text{60 s}}{\text{1 }\cancel{\text{min}}}\right)=1.{\text{4×10}}^{17}\text{s}$.

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{4}\text{.5}\times {\text{10}}^9\text{ years = 1}{\text{.4×10}}^{17}\text{s}$.

c) $\text{71}{\text{.2 cm}}^{\text{3}}$ to m³

Convert the quantity $\text{71}{\text{.2 cm}}^{\text{3}}$ to m³.

Set up the unit conversion calculation using the appropriate conversion factor.

The conversion is as follows:

$\left(\text{71}\text{.2 }\cancel{{\text{cm}}^3}\text{× }\frac{{\text{1 m}}^3}{{\text{10}}^6\cancel{{\text{cm}}^3}}\right)=\text{7}\text{.12}\times {\text{10}}^{-5}{\text{ m}}^3$.

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{71}{\text{.2 cm}}^3=\text{7}\text{.12}\times {\text{10}}^{-5}{\text{ m}}^3$.

d) $88.6{\text{ m}}^3$ to L

Convert the quantity $88.6{\text{ m}}^3$ to L.

Set up the unit conversion calculation using the appropriate conversion factors.

The conversion is as follows:

$\left(\text{88}\text{.6 }\cancel{{\text{m}}^3}\right)\left(\frac{\text{1000 L}}{\text{1 }\cancel{{\text{m}}^3}}\right)=\text{8}\text{.86}\times {\text{10}}^4\text{ L}$.

The units in the conversion factor must cancel to give the correct unit for the answer.

Hence,

$\text{8}{\text{.86 m}}^3=\text{88}\text{.6}\times {\text{10}}^4\text{ L}$.