Problem 3 - College Physics

Question

(a) The recommended daily allowance (RDA) of the trace metal magnesium is 410 \(\mathrm{mg} / \mathrm{day}\) for males. Express this quantity in \(\mu \mathrm{g} / \mathrm{day} .\) (b) For adults, the RDA of the amino acid lysine is 12 \(\mathrm{mg}\) per kg of body weight. How many grams per day should a 75 \(\mathrm{kg}\) adult receive? (c) A typical multivitamin tablet can contain 2.0 \(\mathrm{mg}\) of vitamin \(\mathrm{B}_{2}\) (riboflavin), and the RDA is 0.0030 \(\mathrm{g} / \mathrm{day} .\) How many such tablets should a person take each day to get the proper amount of this vitamin, assuming that he gets none from any other sources? (d) The RDA for the trace element selenium is 0.000070 \(\mathrm{g} /\) day. Express this dose in mg/day.

Step-by-Step Solution

Answer

(a) 410,000 µg/day (b) 0.9 g/day (c) 2 tablets (d) 0.070 mg/day.

Step 1: Convert mg to µg for Magnesium

To convert milligrams (mg) to micrograms (µg), use the conversion factor 1 mg = 1000 µg. Therefore, for magnesium:\[410 \, \mathrm{mg/day} \times 1000 \, \left(\frac{\mu \mathrm{g}}{\mathrm{mg}}\right) = 410,000 \, \mu \mathrm{g/day}\]

Step 2: Convert mg to grams for Lysine

First, calculate the daily number of milligrams a 75 kg adult should consume, using the given rate:\[12 \, \mathrm{mg/kg} \times 75 \, \mathrm{kg} = 900 \, \mathrm{mg}\]Next, convert milligrams to grams:\[900 \, \mathrm{mg} \times \frac{1 \, \mathrm{g}}{1000 \, \mathrm{mg}} = 0.9 \, \mathrm{g}\]

Step 3: Calculate Tablets for Vitamin B2

First, convert the RDA of vitamin B2 from grams to milligrams:\[0.003 \, \mathrm{g/day} \times 1000 \, \left(\frac{\mathrm{mg}}{\mathrm{g}}\right) = 3.0 \, \mathrm{mg/day}\]Now, determine the number of 2.0 mg tablets needed to meet this RDA:\[\frac{3.0 \, \mathrm{mg/day}}{2.0 \, \mathrm{mg/tablet}} = 1.5\]Since you cannot take half a tablet in practical scenarios and to meet at least the RDA, take 2 tablets.

Step 4: Convert grams to mg for Selenium

To convert grams to milligrams, use 1 g = 1000 mg. Therefore, for selenium:\[0.000070 \, \mathrm{g/day} \times 1000 \, \left(\frac{\mathrm{mg}}{\mathrm{g}}\right) = 0.070 \, \mathrm{mg/day}\]

Key Concepts

Conversion of Units

In physics and many other sciences, converting units is a key skill that helps communicate measurements accurately. When dealing with units like milligrams (mg) and micrograms (µg), understanding the conversion factors can make seemingly complex equations straightforward.
For instance, when you need to convert milligrams to micrograms, you use the conversion factor: 1 mg equals 1000 µg. This is because 'micro' signifies a millionth ( 1/1,000,000), and 'milli' signifies a thousandth ( 1/1,000) of a base unit.
This means every milligram contains 1000 micrograms.
This conversion is essential in nutritional science since daily allowances are often listed in different units for precision or convenience.
For example, if the recommended dietary allowance (RDA) of magnesium is 410 mg/day, converting it to micrograms will give you:

• 410 mg/day × 1000 (µg/mg) = 410,000 µg/day

This makes it easier to visualize the small daily requirements for trace elements.

Recommended Dietary Allowance (RDA)

Recommended Dietary Allowances (RDAs) indicate the average daily intake level that is sufficient to meet the nutritional needs of nearly all healthy individuals in a specific life stage and gender group.
RDAs vary from nutrient to nutrient, and for any given nutrient, they can vary over different demographic parameters such as age or gender. They are guidelines set by health organizations to help individuals achieve good nutritional status.
For amino acids like lysine, the RDA is often expressed in relation to body weight, thus allowing for a more precise nutritional prescription.
For a 75 kg adult, if the RDA is 12 mg of lysine per kg, you simply multiply the body weight by this figure to determine how much lysine is needed daily:

• 12 mg/kg × 75 kg = 900 mg/day

Converting this to grams (since nutritional foods may list nutrient content in grams) gives you:

• 900 mg/day × 1 g/1000 mg = 0.9 g/day

These calculations ensure that an individual's dietary intake meets their specific biological needs.

Trace Elements and Biochemistry

Trace elements are minerals required in very small amounts but are crucial for maintaining healthy biochemical and physiological functions in the body. They often act as catalysts in enzyme reactions or are vital components of larger biological structures like proteins or cell membranes.
Selenium is one such trace element. Its RDA is 0.000070 g/day, which might seem minute, but its role in antioxidant enzymes and thyroid function is enormous. To make it easier to interpret, this can be converted into milligrams:

• 0.000070 g/day × 1000 mg/g = 0.070 mg/day

Such conversions make nutritional information more bearable to understand and use.
Vitamins, like riboflavin (Vitamin B2), have their RDA often listed in milligrams. In cases where someone must acquire their nutrients solely from supplements, calculating the number of tablets needed is crucial. If each tablet contains 2 mg of B2 and the RDA is 3.0 mg/day, two tablets ensure adequate intake by exceeding the half-tablet fraction:

• (3 mg/day ÷ 2 mg/tablet) = 1.5; taking 2 tablets ensures RDA is met.

Understanding these basics allows individuals to tailor their nutrient intake, supporting overall health and wellness.