Math 2000 - Solutions 2
Spring 2002
Unit 2A 10b. The area of the pickup's bed is $12 {\rm\ ft} \times 5 {\rm\ ft} = 60 {\rm\ ft}^2$. The bed's volume is $12 {\rm\ ft} \times 5 {\rm\ ft} \times 3.5 {\rm\ ft} = 210 {\rm\ ft}^3$. 12b. To convert 3 miles to feet we use two conversion factors:

\begin{displaymath} 3 {\rm\ miles} \times \frac{1760 {\rm\ yd}}{1 {\rm\ miles}} \times \frac{3 {\rm\ ft}}{1 {\rm\ yd}} = 15,840 {\rm\ ft}. \end{displaymath}

14c. The volume is $50 {\rm\ ft} \times 10 {\rm\ ft} \times 8 {\rm\ ft} = 4000 {\rm\ ft}^3$. To convert this to cubic yards, recall how we find conversion factors for area and volume units. First, we write 3 feet = 1 yard and then raise both sides of this statement to the third power: $(3 \mbox{ feet })^3 = (1 \mbox{ yard })^3$, or 27 feet$^3$ = 1 yard$^3$. Using this conversion factor, we have that 4000 ft$^3$ equals

\begin{displaymath} 4000 {\rm\ ft^3} \times \frac{1 {\rm\ yd^3}}{27 {\rm\ ft^3}} = 148.15 {\rm\ yd^3}. \end{displaymath}

16a. 1 peso is worth $0.1073 (about 11 cents), whereas 1 yen is worth $0.0093 (just under 1 cent), so one peso is worth more. 16d. Note that 4 apples cost 1000 yen, and

\begin{displaymath} 1000 {\rm\ yen} = 1000 {\rm\ yen} \times \frac{ \$0.0093}{1\rm\ yen} =\$9.30. \end{displaymath}

30. If an average human heart beats 60 times per minute, then in 75 years it beats

\begin{displaymath} 60\ \frac{\rm beats}{\rm min} \times 60\ \frac{\rm min}{\rm... ...rm year} \times 75 \ \rm years = 2,365,200,000 \ \rm beats. \end{displaymath}

Unit 2B 6b. A kilometer is 1,000,000,000 times larger than a micrometer. This is because 1 kilometer = 1000 meters, and 1 meter = 1,000,000 micrometers. 8e. Using the conversion factor, 1 km = 0.62 miles, we see that 100 kilometers per hour is the same as

\begin{displaymath} 100\ \frac{\rm km}{\rm hr} \times\frac{0.6214\rm\ mi}{1\rm\ km}= 62.14\ \frac{\rm mi}{\rm hr}. \end{displaymath}

24. The Cullinan Diamond and the Star of Africa. The Cullinan Diamond weighed

\begin{displaymath} 3106{\rm\ carats}\times\frac{0.2\rm\ g}{1\rm\ carat} \times\frac{1000\rm\ mg}{1\rm\ g}=621,200 \rm\ mg. \end{displaymath}

This, in turn, equals

\begin{displaymath} 621,200\rm\ mg \times\frac{1\rm\ g}{1000\rm\ mg} \times\fra... ...\ g} \times\frac{2.205\rm\ lb}{1\rm\ kg}=1.3697 \rm\ pounds. \end{displaymath}

The Star of Africa weighs

\begin{displaymath} 530.2{\rm\ carats}\times\frac{0.2\rm\ g}{1\rm\ carat} \times\frac{1000\rm\ mg}{1\rm\ g}= 106,040\rm\ mg. \end{displaymath}

This, in turn, is

\begin{displaymath} 106,040{\rm\ mg} \times\frac{1\rm\ g}{1000\rm\ mg} \times\... ...rm\ g} \times\frac{2.205\rm\ lb}{1\rm\ kg}= 0.23\rm\ pounds. \end{displaymath}