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nature of mathematics

Questions and Answers of Nature Of Mathematics

Consider selecting five elements, say, \(a,b, c, d\), and \(e\), from the set \(A=\{a,b, c,d, e\}\). List all possible subsets of \(A\) using all five elements. How many arrangements are there?
A typical Social Security number is 555-47-5593. How many Social Security numbers are possible if the first two digits cannot be 0 ?
A club consists of four men and three women. In how many ways can this club elect a president, vice president, and secretary, in that order, if the president must be a woman and the other two
A history teacher gives a 20-question true-false exam. In how many ways can the test be answered if the possible answers are \(\mathrm{T}\) or \(\mathrm{F}\) or possibly to leave the answer blank?
A JVC advertisement appeared in several national periodicals. It claimed that the SEA graphic equalizer system can create 371,293 different sounds from the thirteen possible zone levels on each of
In how many ways can 26 books be arranged on a bookshelf?
The old lady jumping rope in the cartoon at the beginning of Section 12.3 is counting one at a time. Assume that she jumps the rope 50 times per minute and that she jumps 8 hours a day, 5 days a
A marginal note in this section describes the longest paper-link chain. If there are 12 paper links per foot, and it takes 1 minute to construct each link, estimate the length of time necessary to
What numerical property is exhibited by the arrangement of billiard balls shown in Figure 12.3?Figure 12.3 13 3 15 14 10 12 1 9 7 5 8 (103)
"Taste the Rainbow of Fruit Flavors \({ }^{\circledR}\) " Original Skittles bitesized candies come in orange, lemon, lime, grape, and strawberry flavors.Suppose you want to test the number of
How many corks are in the barrel? Outline a procedure that would find the answer without actually counting the number of corks.
A die is a small cube with sides marked with one to six spots on each side arranged so that the opposite sides of the cube have spots adding up to seven. A die can be held so that one, two, or three
The news clip below presents an argument proving that at least two New Yorkers must have exactly the same number of hairs on their heads! For this problem, just suppose you knew the number of hairs
The following transcribed television advertisement is the basis for Problems 56-58.How many different pizza orders are possible? LITTLE CAESAR'S PIZZA PIZZA!!! Customer: So what's this new deal?
a. Jean Sherrod of Little Caesar's Enterprises, Inc., explains that an order where the first pizza is pepperoni and the second pizza is ham would be considered different from an order in which the
Double toppings are not included in the calculation in the advertisement. However, Little Caesar's does allow double toppings. If double toppings are possible, how many different pizza orders are
Show that the sum of the entries of the \(n\)th row of Pascal's triangle is \(2^{n}\).
Prove: \(\left(\begin{array}{l}n \\ r\end{array}ight)=\left(\begin{array}{c}n \\ n-r\end{array}ight)\)
What is the future value after 30 years if you deposit \(\$ 1,000\) and the interest is guaranteed at \(2.5 \%\) compounded continuously?
Suppose you see a car with an advertised price of \(\$ 18,490\) at \(\$ 480\) per month for 5 years. What is the amount of interest paid?
Suppose you see a car with an advertised price of \(\$ 14,500\) at \(\$ 410.83\) per month for 4 years. What is the amount of interest paid?
Suppose you buy a home and finance \(\$ 285,000\) at \(\$ 2,293.17\) per month for 30 years. What is the amount of interest paid?
Suppose you buy a home and finance \(\$ 170,000\) at \(\$ 1,247.40\) per month for 30 years. What is the amount of interest paid?
Find the cost of a home in 30 years, assuming an annual inflation rate of \(1 \%\), if the present value of the house is $ 225,000 .
Find the cost of the monthly rent for a two-bedroom apartment in 30 years, assuming an annual inflation rate of \(10 \%\), if the current rent is \(\$ 850\).
Suppose that an insurance agent offers you a policy that will provide you with a yearly income of \(\$ 50,000\) in 30 years. What is the comparable salary today, assuming an inflation rate of \(6
If a friend tells you she earned \(\$ 5,075\) interest for the year on a 5 -year certificate of deposit paying \(0.5 \%\) simple interest, what is the amount of the deposit?
If John wants to retire with \(\$ 10,000\) per month, how much principal is necessary to generate this amount of monthly income if the interest rate is \(4 \%\) ?
If Melissa wants to retire with \(\$ 50,000\) per month, how much principal is necessary to generate this amount of monthly income if the interest rate is 3%?
In Problems 51-54, calculate the time necessary to achieve an investment goal. Give your answer to the nearest day. Use a 365-day year\(\$ 1,000\) at \(8 \%\) simple interest; deposit \(\$ 750\)
In Problems 51-54, calculate the time necessary to achieve an investment goal. Give your answer to the nearest day. Use a 365-day year\(\$ 3,500\) at \(6 \%\) simple interest; deposit \(\$ 3,000\)
In Problems 51-54, calculate the time necessary to achieve an investment goal. Give your answer to the nearest day. Use a 365-day year\(\$ 5,000\) at \(5 \%\) daily interest; deposit \(\$ 3,500\)
In Problems 51-54, calculate the time necessary to achieve an investment goal. Give your answer to the nearest day. Use a 365-day year\(\$ 5,000\) at \(4.5 \%\) compounded continuously; deposit \(\$
HISTORICAL QUEST In 1981 Home Federal offered an FDIC-insured account that would "quadruple your investment in 10 years." What was the rate of this investment (to the nearest tenth percent) if we
HISTORICAL QUEST On August 8,1981 , American Savings offered an insured tax-free account paying \(23.24 \%\) compounded monthly. If you had invested \(\$ 50,000\) at that time, how much would you
Suppose that \(\$ 1,000\) is invested at \(7 \%\) interest compounded monthly. Use the formula\[ A=P\left(1+\frac{r}{n}ight)^{n t}\]a. How long (to the nearest month) before the value is \(\$ 1,250\)
Suppose that \(\$ 1,000\) is invested at \(5 \%\) interest compounded continuously. Use the formula\[A=P e^{r t}\]a. How long (to the nearest day) before the value is \(\$ 1,250\) ?b. How long (to
What is the subject of Example 14 ? Discuss the real-life application of this example in your own life. Write at least a one-page paper and introduce some new mathematics into your discussion.Data
Consider Table 11.1.Table 11.1a. If this national debt in each of the given years is shared 2009:\$36,000/ equally by the U.S. citizens, how much would it cost person; each of us (rounded to the
In 2009 the world population reached 7 billion. If we assume a growth rate of \(1.5 \%\), when will the population reach 8 billion?
Phoenix, Arizona, had a population of 1,321,045 in 2000 and 1,445,632 in 2010. Predict the population in Phoenix in 2012. Compare this calculated number with the actual 2012 population of
According to the Centers for Disease Control in Atlanta, Georgia, at the end of January 1992, the total number of AIDS-related U.S. deaths for all ages was 209,693. In January 2008 , that number was
If \(100.0 \mathrm{mg}\) of neptunium-239 \(\left({ }^{239} \mathrm{~Np}ight)\) decays to \(73.36 \mathrm{mg}\) after 24 hours, find the value of \(r\) in the growth/decay formula for \(t\) expressed
Carbon-14, used for archaeological dating, has a half-life of 5,730 years. Find the decay rate for carbon-14.
An archaeologist has found a fossil in which the ratio of \({ }^{14} \mathrm{C}\) to \({ }^{12} \mathrm{C}\) is \(20 \%\) of the ratio found in the atmosphere. Approximately how old is the fossil?
A seismograph \(300 \mathrm{~km}\) from the epicenter of an earthquake recorded a maximum amplitude of \(4.9 \times 10^{3} \mu \mathrm{m}\). Find the earthquake's magnitude. What would be the Richter
Compare the strengths of earthquakes with magnitudes 4 and 8 .
Compare the amount of earth movement (the energy released) by earthquakes of magnitudes of 4 and 8 .
The background sound of a study room was measured to be \(10^{-10}\) watts \(/ \mathrm{cm}^{2}\). Find the decibel rating for the room.
The noise in a classroom varies from \(50 \mathrm{~dB}\) to \(62 \mathrm{~dB}\). Find the corresponding variation in intensities.
What is the growth/decay formula?
What do we mean by half-life?
What is a logarithmic scale?
What is a micrometer?
Find the number of decibels for the power of the sound given in Problems 5-10. Round to the nearest decibel.A whisper, \(10^{-13}\) watts \(/ \mathrm{cm}^{2}\)
Find the number of decibels for the power of the sound given in Problems 5-10. Round to the nearest decibel.A traffic jam, \(10^{-8}\) watts \(/ \mathrm{cm}^{2}\)
Find the number of decibels for the power of the sound given in Problems 5-10. Round to the nearest decibel.A rock concert, \(5.23 \times 10^{-6}\) watts \(/ \mathrm{cm}^{2}\)
Find the number of decibels for the power of the sound given in Problems 5-10. Round to the nearest decibel.A conversation, \(3.16 \times 10^{-10}\) watts \(/ \mathrm{cm}^{2}\)
Find the number of decibels for the power of the sound given in Problems 5-10. Round to the nearest decibel.A rocket engine, \(2.53 \times 10^{-5}\) watts \(/ \mathrm{cm}^{2}\)
Find the number of decibels for the power of the sound given in Problems 5-10. Round to the nearest decibel.According to the Guinness Book of World Records, the world's loudest shout, by Skipper
Example 3 calculates the growth rate of AIDS-related U.S. deaths and uses this information to predict the cumulative number of AIDS-related deaths at the end of January 2020. Repeat this example for
Example 3 calculates the growth rate of AIDS-related U.S. deaths and uses this information to predict the cumulative number of AIDS-related deaths at the end of January 2020. Repeat this example for
If the half-life of cesium- 137 is 30 years, find the decay constant,
If the half-life of plutonium-238 is 86 years, find the decay constant,
Find the half-life (to the nearest year) of strontium- 90 if \(r=-0.0246 .\)
Find the half-life (to the nearest year) of krypton if \(r=-0.0641\).
A seismograph \(300 \mathrm{~km}\) from the epicenter of an earthquake recorded a maximum amplitude of \(5.1 \times 10^{2} \mu \mathrm{m}\). Find this earthquake's magnitude.
A seismograph \(20 \mathrm{~km}\) from the epicenter of an earthquake recorded a maximum amplitude of \(8.2 \times 10^{5} \mu \mathrm{m}\). Find this earthquake's magnitude.
Compare the strengths of earthquakes with magnitudes 4 and 6.
Compare the strengths of earthquakes with magnitudes 7 and 8.
The energy released by an earthquake is approximated by\[\log E=11.8+1.5 M\]What is the energy released by the 1906 San Francisco quake, which measured 8.3 on the Richter scale? This energy, it is
The Dead Sea Scrolls were written on parchment at about 100 B.C. What percentage of \({ }^{14} \mathrm{C}\) originally contained in the parchment remained when the scrolls were discovered in 1947 ?
Tests of an artifact discovered at the Debert site in Nova Scotia show that \(28 \%\) of the original \({ }^{14} \mathrm{C}\) is still present. What is the probable age of the artifact?
The half-life of \({ }^{234} \mathrm{U}\), uranium-234, is \(2.52 \times 10^{5} \mathrm{yr}\). If \(97.3 \%\) of the uranium in the original sample is present, what length of time (to the nearest
The half-life of \({ }^{22} \mathrm{Na}\), sodium-22, is \(2.6 \mathrm{yr}\). If \(15.5 \mathrm{~g}\) of an original 100 -g specimen remains, how much time has elapsed (to the nearest year)?
How much more energy was released by the Loma Prieta quake (Richter scale 7.1) than by the Northridge quake (Richter scale 6.6)?
How much more energy was released by the 1960 Chile quake (Richter scale 9.5) than the 1906 San Francisco quake (Richter scale 8.3)?
An artifact was found and tested for its carbon- 14 content. If \(12 \%\) of the original carbon- 14 was still present, what is its probable age (to the nearest 100 years)?
An artifact was found and tested for its carbon-14 content. If \(85 \%\) of the original carbon- 14 was still present, what is its probable age (to the nearest 100 years)?
The radioactive substance neptunium-139 decays to \(73.36 \%\) of its original amount after 24 hours. How long (to the nearest hour) would it take for \(43 \%\) of the original neptunium to be
The 1989 World Series San Francisco quake was initially reported to have magnitude 7.0, but later this was revised to 7.1. How much energy released corresponds to this increase in magnitude?
Compare the amount of earth movement (energy released) by earthquakes of magnitudes 3 and 6 .
Compare the amount of earth movement (energy released) by earthquakes of magnitudes 7 and 8 .
If an object at temperature \(B\) is surrounded by air at temperature \(A\), it will gradually cool so that the temperature \(T\), \(t\) minutes later, is given by Newton's law of cooling:\[T=A+(B-A)
If an object at temperature \(B\) is surrounded by air at temperature \(A\), it will gradually cool so that the temperature \(T\), \(t\) minutes later, is given by Newton's law of cooling:\[T=A+(B-A)
If an object at temperature \(B\) is surrounded by air at temperature \(A\), it will gradually cool so that the temperature \(T\), \(t\) minutes later, is given by Newton's law of cooling:\[T=A+(B-A)
If an object at temperature \(B\) is surrounded by air at temperature \(A\), it will gradually cool so that the temperature \(T\), \(t\) minutes later, is given by Newton's law of cooling:\[T=A+(B-A)
The atmospheric pressure \(P\) in pounds per square inch (psi) is given by\[P=14.7 e^{-0.21 a}\]where \(a\) is the altitude above sea level (in miles). If a city has an atmospheric pressure of
The atmospheric pressure \(P\) in pounds per square inch (psi) is approximated by\[P=14.7 e^{-0.21 a}\]where \(a\) is the altitude above sea level in miles. If the atmospheric pressure of Denver is
The atmospheric pressure \(P\) in pounds per square inch (psi) is approximated by\[P=14.7 e^{-0.21 a}\]where \(a\) is the altitude above sea level in miles. If the pressure gauge in a small plane
A satellite has an initial radioisotope power supply of 50 watts (W). The power output in watts is given by\[P=50 e^{-t / 250}\]where \(t\) is the time in days. Solve for \(t\) to find the time when
A satellite has an initial radioisotope power supply of 50 watts. The power output in watts is given by the equation\[P=50 e^{-t / 250}\]where \(t\) is the time in days. If the satellite will operate
Between 1990 and 2000, the population of St. Louis county grew from 993,529 to \(1,016,315\), but by 2010 had dropped to 998,954. Predict the population of St. Louis county in 2016 using the given
Between 1990 and 2000, the population of Chicago grew from \(2,783,726\) to \(2,896,016\), but by 2010 had dropped to \(2,695,598\). Predict the population of Chicago in 2016 using the given
A bacteria culture had a population of 10 million at 10:00 A.M., and by 2:00 P.M. had grown to 18 million.a. Predict the population at 6:00 P.M. that same day.b. When will the population double in
In October 1999, the world population reached 6 billion.a. If we assume a growth rate of \(0.98 \%\), when did the population reach 7 billion? Answer to the nearest month.b. Compare this with the
IN YOUR OWN WORDS According to the World POPClock, the world population on 7/1/99 was 5,996,215,340 and on 7/1/12 it was 7,023,324,899. Use these numbers to find the annual growth rate for this
A certain artifact is tested by carbon dating and found to contain \(73 \%\) of its original carbon- 14 . As a cross-check, it is also dated using radium, and was found to contain \(32 \%\) of the
The radioactive isotope gallium-67 (symbol \({ }^{67} \mathrm{Ga}\) ), used in the diagnosis of malignant tumors, has a half-life of 46.5 hours. If we start with \(100 \mathrm{mg}\) of \({ }^{67}
IN YOUR OWN WORDS Construct a logarithmic scale. On a sheet of paper, draw a line \(10 \mathrm{in}\). long and mark it in tenths and hundredths. Place the edge of a second sheet along this line and

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