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Honors MAT111 |
Test I – Quantitative / Dr. Kaplan |
Fall 2002 |
1. A warehouse is 40 yards long and 25 yards wide and piled high with cartons to a height of 3 yards.
a. What is the area of the floor in square feet?
(40 yards) x (25 yards) x (9 ft2 / yd2) = 9000 ft2
b. Assuming there is no space between the cartons, what is the total volume of the cartons?
(40 yards) x (25 yards) x (3 yards) = 3000 yd3
(You don’t need to convert to cubic ft, since units aren’t specified.)
c. If 20 percent of the floor is taken up with walkways, what would the total volume of the cartons be?
3000 yd3 - .2 x 3000 yd3 = 2400 yd3, or .8 x 3000 yd3 = 2400 yd3
2. You exercise regularly, so your heart beats at a resting rate of 60 per minute.
a. Assuming that your heart beats at its resting rate 10 hours per day, at 80 beats per minute 10 hours per day while you are going to class and studying, at 100 beats per minute 3 hours a day while you are active, and at 130 beats per minute while you are exercising 1 hour per day. How many times does your heart beat in 10 years?
(60 beats / minute) x (60 minutes / hour) x (10 hours / day) x (365 days / year) x (10 years / 10 yr period) = 131,400,000 beats / 10 yr period
(80 beats / minute) x (60 minutes / hour) x (10 hours / day) x (365 days / year) x (10 years / 10 yr period) = 175,200,000 beats / 10 yr period
(100 beats / minute) x (60 minutes / hour) x (3 hours / day) x (365 days / year) x (10 years / 10 yr period) = 65, 700,000 beats / 10 yr period
(130 beats / minute) x (60 minutes / hour) x (1 hours / day) x (365 days / year) x (10 years / 10 yr period) = 28,470,000 beats / 10 yr period
400,770,000 beats / 10 yr period
b. Repeat this problem with the assumption that you have stopped exercising and your resting pulse is 72 beats per minute and the hour each day you spent exercising is spent sitting with a heart rate of 80 beats per minute. How many times does heart beat in 10 years?
(72 beats / minute) x (60 minutes / hour) x (10 hours / day) x (365 days / year) x (10 years / 10 yr period) = 157,680,000 beats / 10 yr period
(80 beats / minute) x (60 minutes / hour) x (11 hours / day) x (365 days / year) x (10 years / 10 yr period) = 192,720,000 beats / 10 yr period
(100 beats / minute) x (60 minutes / hour) x (3 hours / day) x (365 days / year) x (10 years / 10 yr period) = 65, 700,000 beats / 10 yr period
416,100,000 beats / 10 yr period
3. A solar cell that is 100% efficient generates 1000 watts of power per square meter. Assume that energy from the cells that is not used is stored in batteries and that the cells (counting battery storage losses) are, on average, 15% efficient for 4 hours per day and 10% efficient for 2 hours per day. How many square meters of cells are needed to power the house if the average power requirement is 1 kilowatt?
Energy need = 1 kilowatt x 24 hours = 24 kilowatt hours
15% efficient for 4 hours: (150 watts / m2) x (4 hours) = 600 watt hours / m2 +10% efficient for 2 hours: (100 watt hours / m2) x (2 hours) = 200 watts / m2
= .8 kilowatt hours / m2
(.8 kilowatt hours / m2) x (? m2) = 24 kilowatt hours à 32 m2
4. A toxin finds its way into the water supply and stays at a concentration of 3 x 10-6 for two days before it is discovered and chemically removed. Assuming you have 5L of blood and would develop flu like symptoms if the concentration in your bloodstream reaches .0005%, how much tap water would you need to consume over two days to become sick? Assume that the toxin is not removed from your bloodstream unless you take an antidote. What should health officials advise?
You will become sick when the amount of toxin in your blood reaches (5 L blood) x (.000005 toxin / blood = .000025 L toxin
(? L of tap water) x (.000003 toxin / water) = .000025 L toxin
.000025 L toxin / (.000003 toxin / water) = 8.333 L of tap water
5. If the annual rainfall in York this year is 20 inches and the normal rainfall is 30 inches, express the rainfall this year as a percent of normal and as a percent less than normal.
20/30 x 100 = 66.7% of normal
(20 – 30)/30 x 100 = 33.3% less than normal
6. In 1990, American used 2,626,165 gigawatt (giga = billion) hours of electricity.
a. Express this in watt hours using scientific notation.
2,626,165 gigawatt hours = 2,626,165 x 109 watt hours, since giga means 109
2,626,165 x 109 watts = 2.626165 x 1015 watt hours, since we’ve moved the decimal point six places to the left
b. Find the number of joules (1 watt = joule/s) used in 1990 by the
(1 watt = joule/s) à 1 joule = watt-s
The answer from part a. 2.626165 x 1015 watt hours is number of watts used in 1990
(2.626165 x 1015 watt-hours / year) x (3600 s / hour) = 9.45 x 1018 watt-s / year = 9.45 x 1018 joules / year
7. Estimate the number of words in this textbook by giving a best guess to two significant digits as well as an absolute low and high about which you are sure. Explain how you got your low and your high estimates.
Give a high, low and estimate for the number of words per page, lowering the numbers slightly to take into account pages with few words and blank pages. Then, multiply by the number of pages.
8. If the fusion of the hydrogen in 1 liter of water results 7 x 1015 joules of energy, using the figure you obtained in question 6.b., how many liters of water would supply the
(9.45 x 1018 joules / year) x (1 liter / 7 x 1015 joules) = 1 x 103 liters (only 1 s.d.)
9. How many significant digits are there in the number 0.00005000100?
Rewrite as 5.000100 x 10-5, and then you can count all digits as significant. There are 7 s.d.’s.
10. If you buy 55 kilograms of topsoil, 1.58 kilograms of nutrients for the soil, and a 4 kilogram shovel, what is the weight of your purchase?
55 + 1.58 + 4 = 60.58. You need to round to the unit’s place, since ‘4’ and ‘55’ are only accurate to the unit’s place. The answer is 61.
11. If the CPI index in 1971 was 40.5 and it was 172.2 in 2000
a. What was the cumulative inflation rate over this time period?
The word ‘rate’ is synonymous with relative percent change. (172.2 – 40.5) / 40.5 x 100 = 325% (3 s.d.’s)
b. A dollar in 1971 is worth how much today?
$1 @1971 index x (40.5 1971 index / 172.2 2000 index) = 23.5¢ (3 s.d.’s)
12. If an airline ticket in 1980 cost $230 dollars and it cost $310 dollars in 2000
a. By what percent did the cost of this ticket increase?
($310 - $230) / $230 x 100 = 35% (2 s.d.’s)
b. Is this percent more or less than the CPI increase if the index in 1980 was 82.4?
(172.2 – 82.4) / 82.4 x 100 = 109% (3 s.d.’s)
13. Make up a study that illustrates Simpson’s paradox.
Make the overall number in category A much lower than the overall number in category B; and make the percent associated with the category A higher than the percent associated with the category B.
14. A test for a certain disease is 80% accurate for those who have the disease (true positive) and 80% accurate for those who do not have the disease. The incidence rate for the disease is 1.5% nationally. Below is a table of results from 4000 patients who took the test.
a. Verify that the table below is correct.
60 have disease / 4000 tested x 100 = 1.5% ü
3152 test negative / 3940 no disease x 100 = 80% ü
48 test positive / 60 have disease = 80% ü
a. Of those who have the disease, what percentage test positive?
48 test positive / 60 have disease = 80%
b. Of those who test positive, what percent have the disease? Compare this result to the result in part b. and explain the difference.
48 test positive and have disease / 836 total testing positive x 100 = 5.8% (2 s.d.’s)
c. If you are a doctor using the table below, how would you explain to a patient who tests positive what their chances are of actually having the disease? How does this compare to the national average?
Those who test positive have roughly a 6% chance of having the disease, which while higher than the incidence of the disease also shows that an 80% accuracy rate gives data which is not particularly useful. A doctor should recommend additional tests for those testing positive, especially if other risk factors are present.
|
|
Disease |
No disease |
Total |
|
Test Positive |
48 |
788 |
836 |
|
Test Negative |
12 |
3152 |
3164 |
|
Total |
60 |
3940 |
4000 |