Section 2.4 Dimensional Analysis

Section 2.4 Dimensional Analysis We will need to know a few equivalencies to do the problems in this section. I will give you all equivalencies from this list that you need on the test. There is no need to memorize these formulas. Length 1 in = 2.54 cm 1 ft = 30 cm 1 yd = 0.9 m 1 mi = 1.6 km 1 ft = 12 in 1 yd = 3 ft 1 mi = 5280 ft Volume 1 tsp = 5 ml 1 tbsp = 15 ml 1 fl oz = 30 ml 1 cup = 0.24 l 1 pint = 0.47 l 1 qt = 0.95 l 1 gal = 3.8 l 1 ft3 = 0.03 m3 1 yd3 = 0.76 m3 1 cup = 8 oz 1 pint = 2 cups 1 quart = 2 pints 7.48 gallons = 1 ft3 Area 1 in2 = 6.5 cm2 1 ft2 = 0.09 m2 1 yd2 = 0.8 m2 1 mi2 = 2.6 km2 Weight 1 oz = 28 g 1 lb = 0.45 kg 1 Ton = 0.9 t (tonne) 1 lb = 16 oz 1 ton = 2000 lb Metric Length millimeter 1000 (mm) centimeter 100 (cm) decimeter 10 (dm) meter 1 (m) dekameter 0.1 (dam) hectometer kilometer 0.01 (hm) 0.001 (km) decigram 10 (dg) gram 1 (g) dekagram 0.1 (dag) hectogram 0.01 (hg) kilogram 0.001 (kg) deciliter 10 (dl) liter 1 (l) dekaliter 0.1 (dal) hectoliter 0.01 (hl) kiloliter 0.001 (kl) Metric Weight milligram 1000 (mg) centigram 100 (cg) Metric Volume milliliter 1000 (ml) centiliter 100 (cl) Example: Use dimensional analysis to find the missing quantity. (If you already know how to do this by moving the decimal, you may use that method.) 2000 cm = _____ km I will use the metric length table. I find that 100 cm = 0.001 km Now I will multiply 200 cm by a “unit” with the units I desire in the numerator, and the units I am changing out of in the denominator. 2000cm ∗ 0.001 km 100 cm = 2000∗0.001km 100 = 0.02 km (I cancelled the cm.) Answer: 0.02 km Example: Use dimensional analysis to find the missing quantity. 15 l = _____ cups I will use the top table and I find that: 1 cup = 0.24 l I need to multiply by a unit. Cups will go in the numerator as that is the unit I want to end up with. Liters will go in the denominator as that is the unit I am getting rid of. 1 𝑐𝑢𝑝 15 𝑙 ∗ 0.24 𝑙 = 15∗1 𝑐𝑢𝑝 0.24 Answer: 62.5 cups = 62.5 𝑐𝑢𝑝𝑠 (I cancelled the l.) Example: Use dimensional analysis to find the missing quantity. 90 kg = ______ oz I searched my tables and I could not find a “unit” that compares kg to oz. I will need to use 2 “units” to solve this problem. One unit will convert from kg to lb, and the second will change from lb to oz. These are the units I will use. 1 lb = 0.45 kg 1 lb = 16 oz 1𝑙𝑏 90𝑘𝑔 ∗ 0.45𝑘𝑔 ∗ 16 𝑜𝑧 1𝑙𝑏 = 90∗1∗16 𝑜𝑧 0.45∗1 = 3200 𝑜𝑧 The both the kg and lb’s cancel, as there is one of each in a numerator and a denominator. Answer: 3200 oz #1-30: Use dimensional analysis to find the missing quantity. Round to 2 decimals when appropriate. 1) 10 m = _____cm 2) 500 ml = ____ l 3) 45 mg = _____ g 4) 20 kl = ____ l 5) 5 kl = ______ l 6) 8 kg = _____ g 7) 0.25 kl = ____cl 8) 50 dm = _____ dam 9) 5 ft = ____ in 10) 6 ft = ____ cm 11) 3 mi = ____ ft 12) 4 yd = ____ m 13) 4 l = ____ cups 14) 40 oz = ____ cups 15) 3 fl oz = ____ l 16) 8 fl oz = ____ l 17) 400 oz = ____ kg 18) 6 kg = ____ oz 19) 10000 oz = ____ tons 20) 300 yd = ____ km 21) 21 l = ____ fl oz 22) 6 l = ____ fl oz 23) 10 kg = ____ oz 24) 220 oz = ____ kg 25) 3 tons = ____ oz 26) 2 km = ____ yd 27) 10 cups = ____ quarts 28) 8 quarts = ____ cups 29) 100 tsp = ____ l 30) 2 l = ____ tsp Example: Two cities 1000 miles apart. How many kilometers are they apart? This problem wants me to convert 1000 miles to kilometers. I need to find a unit that does this. 1.6 km = 1 mile 1.6 km 1000 miles ∗ 1 mile = 1000∗1.6 km 1 = 1600 km Answer: The cities are 1600 km apart. Example: Consider the rectangular prism below. a) Find the volume in cubic feet. I will use the volume formula: V = lwh V = (20ft)(5ft)(8ft) Answer: Volume 800 cubic feet b) How many gallons of liquid will it hold? I will use the unit 7.48 gallons = 1 cubic foot. 800 𝑓𝑡 3 ∗ 7.48 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 1 𝑓𝑡 3 = 800 ∗ 7.48 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 = 5984 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 Answer: 5984 gallons Example: A car has an empty 11 gallon gas tank. a) Assume gas is selling for $3.00 per gallon. How much will it cost to fill up the gas tank? This is hardly worth unit analysis. 11 gallons * $3.00 per gallon Answer: $33.00 b) Assume gas is selling for $1.50 per liter. How much will it cost to fill up the gas tank? I need to find out the capacity of the gas tank in liters. Then I can multiply by $1.50 per liter. First convert 11 gallons to liters. I will use 1 gallon = 3.8 l 3.8 𝑙 11 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 ∗ 1 𝑔𝑎𝑙𝑙𝑜𝑛 = 11 ∗ 3.8 𝑙 = 41.8 𝑙 41.8 liters * $1.50 per liter Answer: $62.70 Homework # 31 – 50: Solve, round to 2 decimals when appropriate. 31) The distance from Cincinnati, Ohio to Columbus, Ohio is about 110 miles. What is the distance in kilometers? 32) The distance from Phoenix to Flagstaff is about 140 miles. What is the distance in kilometers? 33) Two cities 400 km apart. How many miles are they apart? 34) Two cities 1000 km apart. How many miles are they apart? 35) A swimming pool holds 12,500 gallons of water. How many cubic feet of water does it hold? (round to 2 decimals) 36) A bath tub holds 50 gallons of water. How many cubic feet of water does it hold? (round to 2 decimals) 37) A pool is 20 ft long, 15 ft wide and 4 ft deep. a) What is the volume of the pool in cubic feet? b) How many gallons of water can it hold? 38) A pool is 25 ft long, 18 ft wide and 5 ft deep. a) What is the volume of the pool in cubic feet? b) How many gallons of water can it hold? 39) Consider the cylinder below. a) What is the volume of the cylinder? b) How many gallons of water can it hold? (Leave your answer in terms of 𝜋. 40) Consider the cone below. a) What is the volume of the cylinder? b) How many gallons of water can it hold? 41) Consider the figure below. a) What is the volume of the shape? b) How many gallons of water can it hold? (use 3.14 for 𝜋 and round your answer to 2 decimals.) 42) Consider the figure below. a) What is the volume of the shape? b) How many gallons of water can it hold? 43) Part of the Queen Elizabeth Way in Canada has a speed limit of 80 kph. What is the speed limit in mph? 44) The speed limit on much of I-17 is 75 miles per hour. What is the speed in kilometers per hour? 45) One gram is the same as 5 carats. David’s new ring contains a precious stone that is 2 carats. Find the weight of the stone in grams. 46) One gram is the same as 5 carats. David’s new ring contains a precious stone that is 1/8 carat. Find the weight of the stone in grams. Answers: 1) 1000 cm 3) 0.045 g 5) 5,000 l 7) 25,000 cl 9) 60 in 11) 15,840 ft 2 13) 16 3 𝑐𝑢𝑝𝑠 15) 0.09 l 17) 11.2 kg 19) .3125 T 21) 700 𝑓𝑙 𝑜𝑧 23) 357.14 oz 25) 96,000 oz 27) 2.5 qt 29) 0.5 l 31) 176 km 33) 250 mi 35) 1671.12 ft3 37a) 1200 ft3 37b) 8976 gallons 39a) 4.608𝜋 ft3 39b) 34.46784𝜋 𝑔𝑎𝑙𝑙𝑜𝑛𝑠 41a) 60840𝜋 𝑓𝑡 3 41b) 1,428,961.25 gallons 43) 50 miles per hour 45) 2/5 gram or 0.40 gram