DIMENSIONAL ANALYSIS #2


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1 DIMENSIONAL ANALYSIS #2 Area is measured in square units, such as square feet or square centimeters. These units can be abbreviated as ft 2 (square feet) and cm 2 (square centimeters). For example, we might say that the area of a room is 328 ft 2. Generally, when you see a unit of measure with an exponent of 2, it is a measurement of area. Similarly, volume is measured in cubic units. The volume of a tank might be written as 200 cubic feet or 200 ft 3. Generally, when you see a unit of measure with an exponent of 3, it is a measurement of volume. The technique of dimensional analysis can be used to convert from one unit of area or volume to another unit of area or volume. However, it is important to remember that more than one dimension is being considered. EXAMPLE 1: Determine the number of square inches in 1 square foot. SOLUTION: This problem is equivalent to solving 1 ft 2 =? in. 2 To solve this problem, remember that 1 square foot means the area of a square that measures 1 foot on each side. The area of the square is A = lw = (1 ft)(1 ft) = 1 ft 2. To convert from square feet to square inches, we must consider that each of the two dimensions of the square is 1 foot, and each foot is equivalent to 12 inches. We can perform the conversion using unit fractions, but we must use the conversion factor of twice, once for each dimension of the square. So there are 144 square inches in 1 square foot. Notice that the unit fraction we used to perform the conversion had 1 foot twice in the denominator (equivalent to 1 ft 2 ) and 12 inches twice in the numerator. REMEMBER: To convert from one unit of area to another unit of area, always use the conversion factor two times. The exponent of 2 on the unit is a reminder that there are two dimensions to consider. 1
2 Performing a conversion between two different units of volume is very similar. Volume involves three dimensions and is expressed in cubic units. For example, 1 cubic meter means the volume of a cube that measures 1 meter by 1 meter by 1 meter. The volume of that cube is given by V = lwh = (1 m)(1 m)(1 m) = 1 m 3. When we perform a conversion using cubic meters, we must remember that there are three dimensions to the cube, each of which measures 1 meter. So the conversion factor must be used three times, as shown in the next example. EXAMPLE 2: Convert 15,800,000 cubic millimeters to cubic meters. SOLUTION: This problem is equivalent to solving 15,800,000 mm 3 =? m 3. We begin by writing 15,800,000 mm 3 as. Now we multiply by a unit fraction that relates mm 3 to m 3 and that has mm 3 in the denominator. From the measurement and conversion table, we know that 1 m = 1000 mm, so we write a unit fraction using the conversion factor of 1000 three times, once for each dimension. So 15,800,000 cubic millimeters is equivalent to cubic meters. The conversion we wrote above with the unit fraction relating mm 3 and m 3 can be written more briefly as shown below. or even as The important thing is to be sure to use the conversion factor three times. REMEMBER: To convert from one unit of volume to another unit of volume, always use the conversion factor three times. The exponent of 3 is a reminder that there are three dimensions to consider. 2
3 We can chain unit fractions to perform conversions with square and cubic units as we did in the earlier problems. The next example illustrates this process. EXAMPLE 3: The displacement of an engine is often measured in liters or in cubic inches. Suppose that a compact car has a 1.8 L engine. What is the displacement of this engine in cubic inches? SOLUTION: This problem is equivalent to the conversion: 1.8 L =? in. 3 Notice only the unit on the righthand side has an exponent of 3. We can compare liters and cubic inches, however, since a liter is a measure of capacity. Remember that 1 ml = 1 cm 3. We will use this relationship to perform the conversion. First we will convert from liters to cubic centimeters by multiplying by two unit fractions. If we multiplied these fractions, liters and milliliters would cancel and we would be left with cubic centimeters. Now we must convert from cubic centimeters to cubic inches, so we multiply by a unit fraction with cubic inches in the numerator and cubic centimeters in the denominator. Multiplying these fractions together gives us the equivalent displacement in cubic inches (rounded to the nearest whole number). Therefore, the displacement of the engine in the compact car is approximately 110 cubic inches. 3
4 PROBLEM SET 2 Answers to the oddnumbered problems are given at the end of the problem set. Dimensional Analysis #2, Continued WARMUP EXERCISES Use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places ft 2 to in in. 2 to cm mm 3 to cm m 3 to L 5. 6 mi 2 to km gal to yd cm 3 to qt acres to yd 2 PROBLEMS Use dimensional analysis to solve each of the following problems. Show the procedure that you used, including all of your unit fractions. Answer the question in a complete sentence. 9. Frank just moved to the U.S. and wants to buy a 40hectare farm. How many acres should he tell the real estate agent he wants to buy? Round your answer to the nearest acre. 10. A room measures 16 ft by 12 ft. a. Find the area of the room in square yards. Round your answer up to the nearest whole number. b. What would it cost to carpet the room if the carpet sells for $24.99 per square yard? Round your answer to the nearest cent. 11. A basement for a 30 ft by 40 ft house is to be dug at a depth of 7 feet. How many cubic yards of earth need to be hauled away? Assume that the surface of the ground is level. Round your answer to the nearest cubic yard. 12. A cylindrical tank is 8 meters long and 4 meters in diameter. How many liters of liquid will the tank hold? Round your answer to the nearest thousand. EXTRA PROBLEMS Use dimensional analysis to perform the following conversions. Show the procedure that you used, including all of your unit fractions. If an answer is not exact, round to two decimal places cm 2 to m yd 3 to ft m 3 to in in. 3 to L Use dimensional analysis to solve each of the following problems. Show the procedure that you used, including all of your unit fractions. Answer the question in a complete sentence. Round your answers to the nearest whole number. 17. Ann s back yard is 60 feet by 70 feet. What is the area of her backyard in square meters? 18. A fish tank measures 25 inches by 16 inches by 12 inches. How many cubic feet of water will it hold? How many gallons will it hold? 4
5 ANSWERS TO ODDNUMBERED PROBLEMS: (NOTE: For some of these problems there are several ways to set up the problem. Therefore, your unit fractions may look different from the sequence of unit fractions shown here. Your final answer, however, should be approximately the same as the one given below.) Frank needs a farm of approximately 99 acres. 11. Volume = (30 ft)(40 ft)(7 ft) = 8400 ft 3 Approximately 311 cubic yards of dirt must be hauled away Area = (60 ft)(70 ft) = 4200 square feet. The area of Ann s back yard is approximately 390 square meters. 5
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