Dimensional analysis is useful for simple unit conversions but if applied properly it is also a generalized problem-solving method. Use the following steps to solve almost any problem involving numbers with units:
Read the problem once through very carefully. Draw a picture if that will help.
Identify and write down all numbers with their units. Include numbers that were written as words and remember to write units such as mph correctly: 25 mph is 25 mi/hr.
Identify the unit required for the answer to the problem.
Set up a conversion process that converts the units of the given numbers into the units of the answer. Make sure to cancel out all other units.
Units can be classified as either length, mass or weight, volume, money, or time. Use separate conversion paths for each type of unit to move toward the units of the answer.
Check your work when you finish. First, check to make sure the units calculation is correct—often students end up with the right unit but it is under the fraction bar. Second, check your arithmetic.
Use dimensional analysis to solve these problems. Write your solutions on a separate piece of paper. Number your work and show each step of your calculation as a careful cancellation of units. Showing your work is critical because the development of skill in communicating what you are thinking is vital to your success in chemistry!
Which is a higher mountain: one with a height of 2,057 m or one with a height of 5,285 ft?
Convert the following times in hours and minutes to decimal hours (for example, 1 hr 30 minutes is 1.5 hours). You convert minutes to hours and then add it to the number of hours given.
1 hr 45 minutes
3 hr 20 minutes
2 hr 15 minutes
5 hr 42 minutes
11 hr 19 minutes
Weights and measures can be tricky. Frequently we use decimals to give parts of a whole, as in 2.75 pounds. But for some weights we use another approach. There are sixteen ounces in a pound so 2.75 pounds is 2 lbs 12 oz. Convert the following decimals to pounds and ounces.
42.125 lbs
3.25 lbs
11.6875 lbs
7.21875 lbs
A pharmacist will often weigh the
medication she is preparing for a patient rather than count all
the individual pills. If she knows that the pills of antibiotic
she is dispensing each have a mass of 0.62 g and she weighs out
18.6 g then how many pills did she dispense?
You and your spouse have just had a baby. The baby is born
while you are studying abroad in a country that uses the metric
system (just about all of them except the US). The nurse tells
you that the baby weighs 3.91 kg and is 51.4 cm long. What is
your baby’s weight in pounds and ounces? What is your
baby’s length in inches?
The distance from the Earth to the Moon is 240,000 miles.
What is this distance in meters?
The peregrine falcon has been measured as traveling up to 350 km/hr in a dive. If this falcon could fly to the Moon at this speed, how many seconds would it take?
The recommended adult dose of Elixophyllin®, a drug used to treat asthma, is 6 mg/kg of body mass. Calculate the dose in milligrams for a 150-lb person.
In March 1989 the Exxon Valdez ran aground and spilled 240,000 barrels of crude petroleum off the coast of Alaska. One barrel of petroleum is equal to 42 gal. How many liters of petroleum were spilled?
The volume of ice in the Greenland Ice Sheet is estimated to be 2,900,000 km^{3}. What is this volume in cubic miles (mi^{3})? In gallons?
Consider the average step made by a long-distance hiker to be 22
inches long. How many steps does a thru-hiker make while
hiking a 575 mile trail?
How many miles can you travel in a car using 13.7 gal of gasoline if your car gets 26 miles to the gallon? How long would it take to drive this distance at an average speed of 53 miles per hour? Give your answer in hours and minutes like this: 1 hr 24 minutes (and not like this: 1.4 hrs).
How many miles could you drive for $11.50 if the gas mileage
of your car is 14 km/liter of gas and the price is $2.95/gal?
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The US quarter has a mass of 5.67 g and is approximately 1.55 mm thick. (1000 mm = 1 m)
How many quarters would have to be stacked to reach 575 ft, the height of the Washington Monument?
How much would this stack weigh in tons? (1 ton = 2,000 lbs)
How much money would this stack contain?
In determining the density of a
rectangular metal bar, a student made the following
measurements: length, 8.53 cm; width, 2.4 cm; height 1.0 cm;
mass 52.706 g. Calculate the density in g/cm^{3}.
Given that gold has a density of 19.3 g/mL what is the mass of a solid gold sculpture with a volume of about 950 mL? What is the weight in pounds?
Consider the gold sculpture in the previous problem. If it was 8 inches tall it could be about 3 inches wide. Considering its small size, would it be wise to toss this sculpture to an unsuspecting friend? Explain.
A student named Murray has been given a piece of metal formed into the shape of a tiger. He has been asked to identify the metal, which is either zinc (d = 7.14 g/cm^{3}) or tin (d = 7.31 g/cm^{3}). The mass of the tiger is 42.0 g. What is the volume of metal in the tiger if it is made of zinc? What is its volume if it is made of tin?
Murray finds that the volume of the tiger is 5.8 cm^{3}. What is the tiger made of? If you can’t tell, then make a suggestion about how Murray could improve his measurement so he can tell the difference between the two metals.
One 1.6 oz. of
package of cinnamon and spice instant oatmeal contains 34 g of
carbohydrates. If you had instant oatmeal 6.0 days a week, how
many ounces of carbohydrate would you consume in a week?
Eggs are shipped from a poultry farm in trucks. The eggs are
packed in cartons of one dozen eggs each; cartons are placed in
crates that hold 20 cartons each. The crates are stacked in the
trucks, 5 crates across, 25 crates deep and 25 crates high. How
many eggs are in 5 truckloads?
David Hill operates a crane that can pick up 3.0 tons of
excavated earth in an hour. Dave’s wages are $35 per hour.
What, then, is the cost of picking up 4,500 kg of excavated earth?