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Dimensional Analysis

Use dimensional analysis to solve the following problems. Show all work. To help you get started always do the following for each problem:

- Read each problem carefully.
- Write down all given and relevant information that you know in the form of conversion factors.
- Figure out what the unit will be for the correct answer and write it down.
- Find a conversion pathway to take you from the units you have to the unit of the answer.
- Check your work.

- Say that $1.44 is equivalent to €1.00 (the euro: the currency used in the European Union). What is $4.50 equivalent to in euros?
- High speed trains in Japan reach speeds of around 200 km/hr. Trains running between Boston and New York can attain a speed of up to 160 mi/hr. Which trains are faster, those in the US or in Japan?
- You pass a road sign stating that the distance to Montreal is 125 km. If you travel at a constant speed of 90. km/hr how long will it take you to reach the city?
- You are in the German city of Freiburg im Breisgau. In the Marktplatz there is a farmer’s market where you find peaches for a price of €2.45 per kilogram. Using the exchange rate quoted in the first problem on this page calculate the cost of a pound of peaches.
- The deepest point in the earth’s oceans is found in the Mariana Trench, a deep crevasse located about 1000 miles south-east of Japan beneath the Pacific Ocean. Its maximum depth is 6033.5 fathoms. One fathom is defined as 6 feet. Calculate the depth of the Mariana Trench in meters.
- You are in the process of choosing a new car. One model you really like gets 33 miles per gallon on the highway. The one your parents like (because it’s safer) gets 18 kilometers per liter. Which car has better gas mileage?

- How many minutes does it take light from the Sun to reach Earth? (The distance from the Sun to Earth is 9.3 × 10
^{7}mi; the speed of light is 3.00 × 10^{8}m/s; Convert mi to m and find a result in s; then convert to min). - Alpha Centauri is the nearest star to our Sun at 4.3 light-years. How far is this in miles? (A light-year is the distance traveled by light in a year, or 365 days; the speed of light is still 3.00 × 10
^{8}m/s. Do a calculation to find out how many miles equals one light-year, then use that as a conversion factor to answer the question.) - The human stomach can expand to hold up to 4.2 quarts of food. A pistachio nut has a volume of about 0.9 mL. Use this information to estimate the maximum number of pistachios that can be eaten in one sitting.
- A pound of coffee yields 50 cups of coffee (4 cups = 1 quart). How many liters of coffee can be brewed using 2.0 kg of coffee?
- On average, water flows over Niagara Falls at a rate of 8.5 × 10
^{4}cubic feet per second. One cubic foot of water weighs 62.4 lb. Calculate the rate of water flow in tons of water per day.

- In the science fiction series called Star Trek the starship U.S.S. Enterprise can travel faster than the speed of light. The various faster-than-light speeds are referred to as warp factors. Warp factor 1.71 is equal to five times the speed of light (3.00 × 10
^{8}m/s). What speed, in mi/hr, is the Enterprise traveling at when it is at warp factor 6? - In water conservation, chemists spread a thin film of an
inert material over the surface of water to cut down the rate
of evaporation of water in reservoirs. This technique was
pioneered by Benjamin Franklin three centuries ago. Franklin
found that 0.10 mL of oil could spread over 40 m
^{2}of water surface. Assuming that the oil forms a*monolayer*, that is, a layer that is only one molecule thick, estimate the length of each oil molecule in nanometers (nm). Hints: 1 mL = 1 cm^{3}, 1 m = 100 cm, 1m^{2}= 10,000 cm^{2}, 1 cm = 1 × 10^{7}nm.