Solutions for Review Sheet Math Q114

Spring 2005

1. a) Mean = (35+20+25+40+60+30+50+75+60+50)/10=44.5 minutes

Median = 45 minutes

b) Histograms will vary depending on intervals chosen.

Here is one example, using intervals that are 15 minutes wide:

c) summary could note that

· Commuting times are diverse – ranging from 20 minute to 75 minutes

· The largest interval was 50 – 64 minutes, representing 40% of the data. The mean and the median are close to this interval.

2. a) Here is the table with relative frequencies inserted. This can be used to make the histogram.

interval	freq.	rel. freq.	representative age
18 - 20	2	17%	19
21 - 23	4	33%	22
24 - 26	1	8%	25
27 - 29	3	25%	28
30 - 32	0	0%	31
33 - 35	2	17%	34
total	12	100%

b) To estimate the mean age, we find an age that represents each interval. These ages are shown in the last column.

Mean age = [19(2)+22(4)+25(1)+28(3)+31(0)+34(2)]/12 = 25.25, or about 25 years.

3. a) number of men between ages of 55 and 60: 26

b) percentage of women between ages of 30 and 40: 28%

c) total number of people aged 60 and over: 26+24=50

Analysis could include the following points:

· The ages for men are skewed to the right, with a peak of 17% in the groups 25 – 30 years old and 30 – 35 years old.

• The ages for women are more uniformly spread out, with a fairly even distribution between ages 20 and 45. More specifically, there were about 12% in the 20 – 25 year group, 13% in the 25 – 30 year group, and about 14% in the 30 – 45 age group.
• For both genders, the ages extended up through 75; it appears that there are a couple of people in the 75 – 80 age group for men, but not for women

4. Looking at the graph, here are some important points to note:

· Population density in central cities is much higher than in other areas, but has decreased since 1950

• Central city population density has decreased from about 7,500 people per square mile to about 3,000 people per square mile – that is, the density has been cut in half.
• Since 1980, population density in the central cities has leveled off at about 3,000 people per square mile.
• It appears that popluation density in metropolitan areas has also decreased since 1950.
• The suburban popluation density has increased since 1950.

Paragraph:

The graph “Population Density by Metropolitan Area Status: 1950 to 2000”, from the U.S. Bureau of the Census, is a dramatic illustration of the change in population in the United States in the last half of the 20^th century. According to the graph, the population density for the central cities in the United States has been reduced by over 50% in the past 50 years. The population dropped from about 7,500 people per square mile inn 1950 to about 3,000 people per square mile in 2000. Much of this drop took place between 1950 and 1980, as the population density has held steady about about 3,000 people per square mile since 1980. The metropolitan areas also showed a decline, but not as dramatic. These areas showed a decrease in population density from about 400 people per square mile to about 300 people per square mile. Historically, this was a time of migration from the central, urban cores to the suburbs. What is interesting to note is that although the population density of suburban areas increased, it did not increase significantly. It appears that as people moved from the urban areas, they moved into suburbs that were much more sparesely populated – the typical population density over this time period in the suburbs is about 100 people per square mile.

5. I would use the median to describe average housing price in Boston. The median represents the “middle value” of an ordered set, and is not influenced by outliers as the mean is. If I used the mean, I might get a skewed figure, since the mean may be pulled upward by the very high incomes of some people in the greater Boston area.

6. a) A function is a relationship between two quantities (usually called x and y), such that each value of x determines exactly one value of y. That is, given one input, there is exactly one output.

b) i) Q is NOT a function of t, since the graph does not pass the vertical line test.

ii) From the table, we see that Weight is NOT a function of Height That’s because when H = 70 in., we have two different values for Weight.

iv) Q is a function of t, since for one particular time, we have exactly one amount of gas in the tank.

7. a) Regression equation is: R = 4.249 – 0.007t

Correlation Coefficient: 0.989

b) The slope is - 0.007 min/year. That means that for every one year increase in time, the record time for the men’s mile decreased by about 0.007 minutes.

c) When would we predict a winning time of 3 minutes? Solve the equation

3 = 4.249 – 0.007t gives t = 178 years, or in the year 2091.

We can use our model to predict the record time for 1965 (= 52 years after 1913):

R = 4.249 – 0.007(52) = 3.885 minutes.

This is pretty close to the actual time of 3.89 minutes.

8. a) Average rate of change in cigarette consumption between 1960 and 1980:

Ave. rate of change = (4.844 x 10¹¹ – 6.31 x 10¹¹)/(1960 – 1980) = 7.35 x 10 ⁹cigarettes per year

Between 1960 and 1997, the average rate of change is

Ave. rate of change = (4.844 x 10¹¹ – 4.80 x 10¹¹)/(1960 - 1997) = 1.12 x 10 ⁸ cigarettes per year

Between 1980 and 1997, the average rate of change is

Ave. rate of change = (6.315 x 10¹¹ – 4.8 x 10¹¹)/(1980 – 1997) = -8.91 x 10 ¹¹ cigarettes per year

b) Add to table:

Year	U.S. Cig. Consumption	U.S. Population	Cigs per person
1960	484,400,000,000	180,000,000	2691
1970	536,400,000,000	204,000,000	2629
1980	631,500,000,000	227,200,000	2779
1990	525,000,000,000	249,400,000	2105
1997	480,000,000,000	267,800,000	1792

To obtain the numbers in the fourth column: take cigarette consumption and divide it by the U.S. population.

c) Between 1960 and 1997, rate of change in number of cigarettes smoked per person is:

(1792 – 2691)/37 = -24.2 cigarettes per person.

d) Although the number of cigarette smokers in 1997 is about the same as that number from 1960, we can’t conclude that smoking was as popular in 1997 as it was in 1960. We need to look at the percentage of Americans who smoked, not the raw numbers. The size of the population grew in that time period, and our work with column 4 in the table shows that the number of cigarettes per person decreased during this time. Thus, we conclude that smoking is not as popular in 1997 as it was in 1960.

9. C₁ = 0.15t

C₂ = 4.95 + 0.10 t

10. a) (3.54 x 10²¹)(8.9 x 10^-66) = (3.54)(8.9) x 10 ^{21 – 66} = 31.51 x 10 ^–45 = 3.151 x 10 ^–44.

b) (7.9 x 10⁸⁹) / (1.33 x 10⁵⁵) = (7.9/1.33) x 10 ^{89 – 55} = 5.9 x 10^34.

c) (6.7 x 10 ⁷³) ³³ = 6.7 ³³ x (10 ⁷³)³³ = (1.8 x 10 ²⁷)(10 ²⁴⁰⁹) = 1.8 x 10²⁴³⁶.

11. a) For computer company Alpha, slope is 29.99 $ per month and initial value is $399. Linear equation is: a = 399 + 29.99m, where a = total cost and m = number of months

b) independent = m (number of months)

dependent = A ($, total cost for Alpha plan)

slope = 29.99 $/month

vertical intercept = $399

c) The domain refers to the set of values for the independent variable (months) that make sense. In this case, it is reasonable to use m between 0 and 60 months, as most computers last about 5 years before being replaced.

d) Beta: slope = 49.99 $/month, but initial value is $0.

i) Equation is b = 49.99m, where b = total cost for Beta plan and m = number of months.

ii) Use Excel to create table and graph:

e) From the table and the graph, we see that for the first 20 months, Alpha is the cheaper plan. After that, Beta is cheaper. Whether you would use plan Alpha or Beta depends on several factors. With plan Alpha, you own the computer and, after 24 months, could switch to a different (maybe cheaper) internet service. With plan Beta, you may be able to switch to a different plan and upgrade to a different computer after awhile.

12. We know that 1 km = 0.62 miles. Then

58 miles 1 km = 93.55 km per hour.

1 hour 0.62 miles

Since there are 60 minutes in 1 hour, the speed is 93.55 km/hour x 1 hr/60 min = 1.55 km per minute.

13. 1 Earth diameter = 2 x 10⁷ meters; 1 hydrogen diameter = 5.29 x 10 ^–11 meters.

Then 1 Earth diameter = 2 x 10⁷ meters x 1 hydrogen diameter = 3.8 x 10¹⁷

5.29 x 10 ^–11 meters

14. Diameter of earth is 6.2 x 10⁶ meters, which we round to 10⁷. Thus we estimate that the sun’s diameter is about 10¹⁰ meters.

15. The CEO’s compensation is 7.85x10⁸, which we round to 10⁹. The average worker earns about 2.8x10⁴, which we round to 10⁴. Thus, the CEO’s earnings are 5 orders of magnitude higher than the average worker’s.

16. 13% of 2.4x10¹² = 0.312 x 10¹² or 312 billion dollars.

17. To calculate the amount per person, find the ratio of $ to people. That is, 1.4x10⁹/6.4x10⁶ = $218 per person.

18. a) Variables: independent is t, representing years since 1998; dependent P is population of China (billions).

b) linear model: P = 1.237 + 0.03t

c) exponential model: P = 1.237(1.024)^t

d, e) graph and spreadsheet:

f) In year 2050, t = 52. Using the linear model, P = 1.237 + 0.03(52) = 2.797 billion.

Using the exponential model, P = 1.237(1.024)⁵² = 4.25 billion.

19. a) half-life is 30 years: this means that in 30 years, one half of the original population has decayed.

b) If initial population is 180 grams, then after 30 years we have 90 grams. After 60 years we have 45 grams, and after 90 years we have 22.5 grams.

c) Exponential equation: P = 180 (0.5)^t/30 = 180 (0.977)^t.

21. a) y = 817.17e^0.0091x

b) Domain is 0 ≤ x ≤ 200; Range is 980 ≤ y ≤ 6060

c) Growth factor = e^0.0091 = 1.0091

d) The percentage growth rate is 0.91%, or 0.0091

e) In 2050, x = 250. Population is: 817.17(e^0.0091x)²⁵⁰=7949, or about 7950 million.

21. As you read the article it seems that the time period being discussed is 1996 through 2001. During this time, the number of people on welfare dropped from 12.2 million to 5.2 million, a decrease of 7 million welfare recipients.

a) 12.2 million people were on welfare in August, 1996.

b) Calculate the rate of change as change in people on welfare divided by change in years:

(12.2-5.2)/(1996-2001) = 7/(-5) or -1.4 million people per year.

c) Let x = years since 1996

Let Y = number of people on welfare (in millions)

Slope = -1.4 million people per year

Initial value (y-intercept) =12.2 million people

function is Y = 12.2-1.4x

so if t= 7 (that is, 7 years since 1996, or the year 2003) then Y = 12.2= 1.4(7) = 4.92 million people on welfare.