One must learn by doing the thing; for though you think you know it, you have no certainty until you try.
Sophocles

DIMENSIONAL ANALYSIS PROBLEMS

1. We are moving away from some of our oldest units of weight and measures. But if your elderly physician said, "Nothing to worry about. Just take two and a half scruples of aspirin and call me in the morning.", you had better be able to determine the number of aspirin tables you will need to swallow! How many aspirin tablets will you take if your over the counter tablets contain five grains of aspirin?
ans:
10 aspirin tablets

2. A sample of gold has a mass of 1.245 kilograms. The price of gold is currently priced on the New York futures market at $6.41 per troy ounce. What is the value of the gold?
ans. $ 256.58

3. A developer buys a some land that contains a woods. The property measures 100 rods by 50 chains. The developer paid $1,500,000 for this parcel of land. What was the cost per acre?

ans: $12,000 per acre

4. The price of gasoline this morning was $1.99 We are planning a trip out east to visit our daughter, and we estimate the total distance roundtrip to be 2480 miles. What will the gasoline cost for the trip?The owners manual stated that our car gets 15 Km per liter.

ans. $137

5. It is known that four hundred pounds of iron metal occupy a volume of 0.02234 cubic meters. Calculate the cubic feet of iron in a statue that has been determined to contain 54.5 kilograms of iron.
ans. 0.23 ft3 of Fe
6. In the fall of 1999, 62 students enrolled in Dr. Fuller's CHEM 151 class at the College of DuPage. Thirty-two percent of the students originally enrolled withdrew from his class before the end of the quarter. Of the remaining students, 26% received A's and 16% received B's. How many students received A's?

ans. 11 students received A's

7. The density of carbon tetrachloride is 1.60 grams of carbon tetrachloride per cc of carbon tetrachloride. What is its density expressed in tons per cubic yard?
ans. 1.35 Ton CCl4 per cubic yard

8. Olive oil has a density of 0.89 grams per mL. If a recipe calls for nine ounces of olive oil, how many cups of olive oil are required in the recipe?
ans. 1.2 cups of olive oil

9. Bromine is used to make flame retardant plastics and cloth and to make photographic film. The bromine used in making these and other products is obtained from concentrated brines. Natural brines in Michigan and the Dead Sea contain up to 0.5% bromide (by mass) . Seawater contains 0.0065 % bromide, the density of seawater is found to be 1.025 g/mL. How many gallons of seawater will contain 600 pounds of bromide, Br-?

ans. 1.05 x 106 gallons

10. Suppose someone offered to give you a 2 gallon bucket filled with gold if you carry the bucket of gold up a flight of stairs. But if you fail to carry the bucket up the flight of stairs, you must stay in CHEM 151 forever. Would you accept the offer? The density of gold is 19.3 g/cc.
ans. 320 pounds of gold
11. Vanadium metal is added to steel to impart strength to the metal. Vanadium has a density of 5.96 g/cm3. The metallurgist adds 580 pounds of vanadium of a batch of steel. What volume, expressed in cubic feet, of vanadium was added to the batch of steel?

ans. 1.56 cubic feet

12. How many rundlets are there in 0.25 ton of acetylsalicylic acid? In the land of Yor, the density of acetylsalicylic acid is reported to be 1.28 scruples per mL.
ans. 2.0 Rundels

13. The futures market on May 19, 2003 listed the opening price of unleaded gasoline at $0.5462 per gallon. What is the value of a storage tank filled with unleaded gasoline? The cylinder storage tank is 35 feet tall and 45 feet in diameter.
ans. $ 2.3 x 105

14. A sample of milk is found to have arsenic at a concentration of 1.32 x 103 micrograms/mL. What are the concentration in ounces per gallon?
ans. 0.00076

15. An average man requires about 200mg of riboflavin (vitamin B2) per day. How many tablespoons of cheese would a man have to eat each day if this was the only source of riboflavin and if mozzarella cheese contained 5.5 mg. of riboflavin per gram? The density of mozzarella cheese is 0.6893 grams per mL.
ans.
3.5

16.  If one gram of silver can be converted into 400 square feet of mirrors, how thick is the coating.  The density of silver is 10.5 grams per cm3.
ans.
3.18 x 10-8

17. A child is to be given 1 teaspoon of Amoxil 3 times a day for twelve days. Amoxil is a semi-synthetic antibiotic with a broad spectrum of bactericidal activity against both gram-positive and gram-negative bacteria. Amoxil used orally and comes in bottles with either 100, 150, 225, or 250 mL of Amoxil. As the physician, you will write the prescription to supply sufficient Amoxil but no refills. What quantity will you prescribe and what size bottle should be used?
ans. 225 mL bottle

18. One serving of Frank's Quality Kraut Juice supplies 1800 MG of sodium. A single serving of Frank's Quality Kraut Juice weighs 7.5 ounces. 100% of a persons daily value of sodium is equivalent to 2,400 MG of sodium this value is based on a 2,000 calorie diet. How many cups of Frank's Quality Kraut Juice would need to be consumed to supply 1500 MG of sodium. The density of Frank's Quality Kraut Juice is 1.5 grams/mL.
ans. 3/4 of a Cup

19.  Dr. Fuller does not drive over the speed limit 97.3% of the time. He figures he has about a 4% chance of being stopped if he is speeding and only about a 15% probability of not being given a ticket when stoped by the police. For the last five years he has averaged about 1100 hours a year behind the wheel of his sports car, an old Toyota 1993 Paseo. Since a speeding ticket cost roughly $95 dollars. How much should he budget for tickets each year?
ans.   $95

20. An "acre foot" of water will cover an acre of land with a layer of water one foot deep. How many tons are in a six "acre foot" of water?
ans. 8000 tons

21. In the settling of south central Louisiana, Cajun country, the principal means of transportation was by waterway. It was necessary that all settlers have access to the waterway. In the 1700's, the French measurement for land area was the arpent. The arpent is still used today in French speaking sections of Canada and Louisiana. A Cajun would customarily own about tw0 arpents of wetland with about sixty feet on the bayou. If the property was laid out in rectangular lots, what was the length of the property? What is a bayou, an arpent, an ares?
ans. 1000 feet

22. The estimated quantity of recoverable crude oil form the field at Prudoe Bay in Alaska is 1.53 x 109 cubic meters. What is this currnet potential value of the estimated crude oil? The density of Prudoe Bay crude oil is 0.894 g/mL and crude oil is presently selling for $35.89 a barrel.

ans. 3.45 x 1011

23. If a turtle travels 20 centimeters in one minute, how many furlongs would it travel in a fortnight? Assume that the turtle rests 8 hours each day.
ans. 13 furlongs per fortnight

24.   ..."Excessive use of saccharin by anyone is ill-advised and should be actively discouraged." This statement appeared in the New England Journal of Medicine. It stated excessive use of saccharin is generally considered to be drinking four or more artificially sweetened beverages per day. Since a drink contains 80 MG of saccharin, how many 1/4 grain tablets of saccharine would be equivalent to the four drinks?
ans. 22 Tablets

25.  The American Cancer Society a few years ago ordered 40 billion units of interferon, at a cost of $2 million dollars, to use in studies on six types of cancer in 150 patients. A unit of interferon on six types of cancer in 150 patients. A unit of interferon weighs one picogram. That year 100 MG of interferon was produced from 31,000 liters of blood. At that rate production, how many liters of blood did it take to supply the order for the A.C.S.?
ans. 12,400 Liters of Blood

26. If one troy ounce of silver can be converted into 400 square feet of mirrors, how thick is the coating? The density of silver 10.5 grams per mL.
ans. 8x10-5 mm

27. Commercial solvents are commonly sold to merchants in 55-gallon drums. What is the value of a 55-gallon drum, if the carbon tetrachloride can be resold for $2.95 for a 250 mL bottle? Carbon tetrachloride is 1.59 times as dense as water.
ans. $ 2456 per drum of carbon tetrachloride

28.  The diameter of metal wire is often referred to by its American wire gauge number. A 20-gauge wire has a diameter of 0.03196 inches. How many meters of wire are present in a 3.0 pound spool of 20-gauge copper wire? The density of copper is 8.92 g per mL.
ans. 300 meters in a 3 pound spool

29. We are planning a trip out east to visit our daughter. The price of gasoline this morning was $1.999, and we estimate the total distance round trip to be 2480 miles. What will the gasoline cost for the trip? The owner's manual stated that our car should average 15 km per liter.

ans. $110.69

30. OSHA has set an upper limit of 5 PPM, parts per million, of benzene, C6H6, to be permitted in the atmosphere. Oxygenated gasoline at one time contained 20% by volume benzene. How many mL of gasoline would have to evaporate in a room to produce 5 ppm? Assume the room is 14 concrete blocks high and the floor is 59 tile long by 32 tile wide. Benzene is a colorless liquid which has a density of 0.87845 gram per milliliter. Information: ppm is equivalent to 1 MG per 1000 liter of air; tiles are twelve inch squares; concrete blocks are about 8 inches high.
ans. 14.5-mL ( 0.987 ~ 1 tablespoon) of gasoline

31. A 1957 Chevy automobile averages 16.0 miles per gallon of gasoline. If a drop is a sphere of radius 2.3 mm, how many drops of gasoline are needed in order to travel 1.25 kilometers?
ans. 11300 drops

32. This last spring I put an emergency submersible pump in our basement. To prepare the dry lead storage battery for use, I bought some battery acid to be used for lead storage batteries. The battery acid was 38% sulfuric acid and 62% water by mass according to the label on the plastic container. How many grams of sulfuric acid were in the 2 quart container of battery acid? The density of the battery acid was given as 1.285 g/mL.
ans. 950 grams of sulfuric acid

33. The Ardabil carpet was made in 1519-1540. The carpet is a British treasure and hangs in the Victoria and Albert Museum in London. The carpet is 10.5 meters by 5.33 meters and has 340 Persian knots per square inch. If a child worker could average tying 2 knots each minute, how many working days would be necessary to make this carpet? There were no child working laws in the 1500's and the children were worked 11 hours per day. The carpet was supposedly completed in 415 days; how many workers would this have required?
ans. 27 workers

34. In late August 1996 people in Detroit traveled to Windsor, Canada to buy gasoline. How much did a Detroit motorist save (in US dollars) when the motorist bought 15 gallons (US) of gasoline? Unleaded gasoline sells for 30.3 cents (Can.)/liter in Canada and $1.299 (US)/gallon in Detroit. The motorist drove an extra 4 miles each way to buy the unleaded gasoline. Assume that the motorists car gets 19 miles per gallon for this city driving. It costs $1.00 in tolls each way to cross the river, payable on the side of origin in their currency. The rate of monetary exchange was 21.00 equivalent to $1.28 (Can.).

35. During the sweet corn season in 1988, the Fullers proposed freezing the sweet corn obtained from six bushels of sweet corn, eating the corn from an additional two bushels, and giving three more bushel of sweet corn to their friends. The length of the garden could only be sixty feet wide because they were raising hay that year in the adjacent field. Sweet corn produces two ears of sweet corn per plant but the variety used, Illini Extra Sweet, did so only about 80 percent of the time. The corn planter, their son, manually placed the seeds 16 inches apart. The rows were 36 inches apart in order to use the rototiller for cultivating the soil. An average bushel of sweet corn usually contains around six dozen ears. How many 60 foot rows of corn were planted and what width had to be set aside for planting and growing the sweet corn?

36. Cabot Chemical's chemical plant in Franklin, Louisiana converts natural gas to carbon black. Carbon black is used as an ingredient in plastics, automobile tires, paints, inks, etc. Last year, the Franklin plant had the capacity to produce 230 million pounds of carbon black. It takes 16 tons of natural gas to produce 12 tons of carbon black. Tuesdays bid price for natural gas on the New York Mercantile exchange was $1,595 for 1 million BTU. Natural gas supplies 1000 BTU's per cubic foot. At 60oFand one atmosphere pressure, 1000 cubic feet of natural gas weights 42.25 pounds. If the plant is running at 50% of capacity, what investment in natural gas would have to be made to supply sufficient natural gas for one year?

37.TIME, October 28, 1991, page 79. A paragraph taken from a three page article in TIME (This question has been assigned since November 1991, long before George Bush Jr. became President of the United States or was even the Gov. of Texas. )

George Bush Jr. was probably never a prime mover behind the Bahrain deal. In fact, board members say he voiced doubts about whether Harken had the means and expertise for such a distant oil play. Even so, he has already earned a handsome profit from it. In late June 1990, five months after the deal was sealed and about a month before Iraq invaded Kuwait, young Bush sold 66% of his Harken stake (or 212,140 shares) at the top of the market for nearly $850,000, which represented a 200% profit on his original stake. What was his original cash investment in Harken Energy Corp.? What was the total number of shares purchased and their purchased price per share? What was the value of the stock per share five months after he bought the stock? What was the total net worth in late June 1990?30. Given the LDLo for humans, calculate the oral dose required for a person whose weight is 160 pounds.

38. It is a common practice to draw correlations between toxicity data for test animals and humans. The LD50 ** for oral - ingestion is given below in problem 32. If you had a 160 pound guinea pig (human, how many grams of BaCl2 would be required to produce a 50% kill?

39. Given the MSDS information, calculate the time (hours) of exposure to BaCl2 at the maximum allowed exposure of BaCl2 administered orally? Assume your tidal volume*** is 500 mL and the average number of inspirations is 16 inspirations per minute [inspirations vary form 12 - 20 per minute].

The MSDS for BaCl2 lists:
Toxicity data: 11,400 microgram/KG LDLo* oral-humans;
70 mg/KG LD50** oral-guinea pigs;
55 mg/KG subcutaneous-guinea pig LDLo Irritant;
250 mg(Ba)/cubic meter immediately dangerous to life or heath. Acute exposure may cause irritation of the respiratory tract with sore throat, coughing, and labored breathing. If sufficient amounts are absorbed, symptoms may resemble those of acute ingestion.... No data available;
Mutagenic data: reproductive effects data.
Carcinogen status: None
Disposal: Regulatory level: only above 100.0 MG/liter
*LDLo: the lowest dose of a substance introduced by any route other than inhalation, over any given period of time in one or more divided portions and reported to have caused death.
**LD50: a calculated dose of a substance that is expected to cause the death of 50% of an experimental animal population. It is determined from the exposure to the substage by any route other than inhalation of significant number from the population.
***Tidal volume; the volume of air inspired or expired during a normal inspiration.
ans. 29 hours and 22 minutes

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