Week 2: Dimensional Analysis To Solve

Week 2 Dimensional Analysisuse Dimensional Analysis To Solve The Follo

Use dimensional analysis to solve the following word problems: 1. According to the Guinness Book of Records the heaviest baby ever born weighed 29 lbs 4oz (29.25lbs). What was the baby’s mass in grams? Historical Note: The birth occurred in Effingham IL in 1939 and due to respiratory problems the baby died two hours later. The heaviest babies to survive weighed 22.5 lbs and were born in 1955 and 1982. (16oz. = 1 lb = 454g) 2. In Europe gasoline is sold by the liter. Assume that it takes 14 gallons of gasoline to fill the tank of a compact car. How many liters of gasoline will it take? ( 1L = 0.264gal) 3. The food that the average American eats in one day provides 2500 Calories of energy. How many Calories per second is this? 4. The Atlantic Ocean is growing wider by about 0.10 inch/year. There are 12 inches/ft and 5280-ft/ mile. How long will it take for the Atlantic to grow 1 meter? (1m = 3.23ft) 5. Cheetahs are the fastest land mammals and are capable of sprinting at 27.8 m/s in short bursts. How long would it take a cheetah to run the length of a 100 yd football field running at this top speed (1 yd = 0.914 m)? 6. If Gasp cigarettes have 5.0 mg tar and 0.40 mg nicotine per cigarette and there are 20 cigarettes per pack, how many packs of cigarettes would have to be smoked to coat your lungs with 8.0 oz (1/2 lb.) of tar? (16oz. = 1 lb = 454g) 7. The physician has order 0.650 g of oxycodone to be given to the patient ever 8 hours. The pharmacy has 325 mg tablets of oxycodone, how many tablets will be needed for a week’s treatment? 8. The doctor asks for an infusion of procainamide at a rate of 2.5 μg/min. The pharmacy has mixed 3.0 g of procainamide in 1.0 L of solution. How many mL/hr would you set the IV pump? 9. The physician orders IV fluids to hydrate a client. The order is written as "D5NS 4 Liters over 24 hours." Looking at the package indicates the drip factor of the tubing is 15 gtt/mL (drops/mL). What is the drip rate in gtt/min? 10. If the RDA for vitamin C is 60.0 mg per day and there are 70.0 mg of vitamin C per 100.0 g of oranges, how many 3.0 oz. oranges would you have to eat each week to meet this requirement?

Paper For Above instruction

Dimensional analysis is a vital mathematical tool in physics and healthcare, enabling practitioners and students to convert units, compare quantities, and solve real-world problems accurately. By applying unit conversion factors systematically, complex problems become manageable, ensuring precision in calculations that illustrate principles such as growth rates, dosages, and physical measurements. The following solutions demonstrate the application of dimensional analysis to a variety of word problems across different contexts, emphasizing the importance of accurate unit conversions and mathematical reasoning.

Problem 1: Baby’s mass in grams

The heaviest baby ever born weighed 29 lbs 4 oz. First, convert ounces to pounds: 4 oz ÷ 16 oz/lb = 0.25 lb, so total weight is 29 + 0.25 = 29.25 lbs. To convert pounds to grams, utilize the conversion factor 1 lb = 454 g: 29.25 lbs × 454 g/lb ≈ 13,270.5 g. Thus, the baby’s mass was approximately 13,271 grams. This calculation underscores the importance of unit conversions in understanding biological weights and their conversions to metric units.

Problem 2: Liters of gasoline to fill a tank

Given that 14 gallons of gasoline are needed to fill the tank, and knowing 1 L = 0.264 gal, convert gallons to liters: 14 gal ÷ 0.264 gal/L ≈ 53.03 L. Therefore, approximately 53.03 liters of gasoline are required. This illustrates the use of conversion factors in fuel estimation and automotive planning.

Problem 3: Calories per second

The daily caloric intake is 2500 Calories. To find the rate per second, divide by the total seconds in a day: 24 hours × 3600 sec/hour = 86,400 sec. Then, 2500 Calories ÷ 86,400 sec ≈ 0.0289 Calories/sec. This calculation highlights the importance of time-based units in nutritional sciences and metabolism studies.

Problem 4: Growth of the Atlantic Ocean

The ocean widens by 0.10 inch/year. Convert inches to feet: 0.10 inch ÷ 12 inches/ft ≈ 0.00833 ft/year. Next, determine how many years it takes to grow 1 meter (3.23 ft): 3.23 ft ÷ 0.00833 ft/year ≈ 387.7 years. This demonstrates the use of dimensional analysis in geological timeframes and continental drift studies.

Problem 5: Cheetah’s running length of a football field

The cheetah sprints at 27.8 m/sec. The length of a football field is 100 yards, which converts to meters as 100 yards × 0.914 m/yard = 91.4 meters. Time = Distance ÷ Speed: 91.4 m ÷ 27.8 m/sec ≈ 3.29 seconds. This illustrates the application of motion equations and unit conversions in biological and sports sciences.

Problem 6: Cigarettes needed to coat lungs with tar

Each cigarette contains 5.0 mg tar. To coat lungs with 8 oz (half a pound) of tar, convert 8 oz to grams: 8 oz ÷ 16 oz/lb × 454 g/lb ≈ 226.75 g of tar. Convert grams to milligrams: 226.75 g × 1000 mg/g ≈ 226,750 mg. Number of cigarettes = total mg of tar ÷ tar per cigarette = 226,750 mg ÷ 5 mg ≈ 45,350 cigarettes. Since each pack has 20 cigarettes, the number of packs = 45,350 ÷ 20 ≈ 2267.5 packs. This calculation emphasizes large-scale exposure assessments in health sciences.

Problem 7: Oxycodone tablets for a week’s treatment

Dosage required is 0.650 g every 8 hours. Weekly dose = 0.650 g × 3 doses/day × 7 days ≈ 13.65 g. Tablets each contain 325 mg = 0.325 g. Number of tablets = total dose ÷ dose per tablet = 13.65 g ÷ 0.325 g ≈ 42 tablets. This showcases medication dosing calculations, critical in pharmacy practice.

Problem 8: Infusion rate of procainamide

The ordered dose is 2.5 μg/min. The solution concentration is 3.0 g in 1.0 L, so the concentration is 3.000 g / 1000 mL = 0.003 g/mL = 3 mg/mL. The required infusion rate in mL/min is (2.5 μg/min) ÷ (3000 μg/mL) ≈ 0.000833 mL/min. Convert to mL/hr: 0.000833 mL/min × 60 min/hr ≈ 0.050 mL/hr. This precise calculation is vital in IV medication administration.

Problem 9: Drip rate in gtt/min

The order is 4 liters over 24 hours. Convert liters to mL: 4 L × 1000 mL/L = 4000 mL. The drip factor is 15 gtt/mL, so total drops = 4000 mL × 15 gtt/mL = 60,000 gtt. Time in minutes = 24 hours × 60 min/hour = 1440 min. Drip rate = 60,000 gtt ÷ 1440 min ≈ 41.67 gtt/min. This calculation ensures accurate fluid delivery in clinical settings.

Problem 10: Oranges to meet vitamin C RDA

The recommended daily intake is 60 mg. Each orange provides 70 mg per 100 g. To find how many oranges are needed weekly: daily requirement per day is 60 mg, so weekly intake = 60 mg × 7 ≈ 420 mg. Each orange weighs 3.0 oz. Convert ounces to grams: 3.0 oz ÷ 16 oz/lb × 454 g/lb ≈ 85 g. Vitamin C per orange = (70 mg/100 g) × 85 g ≈ 59.5 mg. Number of oranges per day ≈ 60 mg ÷ 59.5 mg ≈ 1.01 orange. Weekly oranges = 1.01 × 7 ≈ 7 oranges. This calculation guides dietary planning for nutrient intake.

References

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