Question

One cup of Raisin Bran provides \(60 . \%\) of the U.S. recommended daily allowance (RDA) of iron. (a) If the cereal provides \(11 \mathrm{mg}\) of iron, what is the U.S. RDA for \(\mathrm{Fe}\) ? (b) When the iron in the cereal is extracted, it is found to be the pure element. What is the volume of iron in a cup of the cereal?

Short answer

The U.S. RDA for iron is approximately 18.33 mg. The volume of iron in a cup of Raisin Bran is approximately 0.0014 cm³.

Step-by-step solution

Identify Given Information

The problem gives the following information: - One cup of Raisin Bran provides 60% of the U.S. RDA of iron.- This corresponds to 11 mg of iron.We need to determine the total U.S. RDA for iron.

Set Up the Equation

Let the total U.S. RDA for iron be \text{RDA}We know that 60% of this RDA equals 11 mg of iron.\[ \text{0.60} \times \text{RDA} = 11 \text{ mg} \]

Solve for the U.S. RDA

To find the RDA, we can rearrange the equation:\[ \text{RDA} = \frac{11 \text{ mg}}{0.60} = 18.33 \text{ mg} \]So, the total U.S. RDA for iron is approximately 18.33 mg.

Convert 11 mg to Grams

Since we need the volume of the pure iron and density is generally given in g/cm^3, convert 11 mg to grams:\[ 11 \text{ mg} = 0.011 \text{ g}\]

Use Density to Find Volume

The density of iron is given as 7.874 g/cm\text{³}, use the formula\[ \text{density} = \frac{\text{mass}}{\text{volume}}\] Rearrange to find volume:\[ \text{volume} = \frac{\text{mass}}{\text{density}} = \frac{0.011 \text{ g}}{7.874 \text{ g/cm}³} \] \[ \text{volume} \text{ = approximately 0.0014 cm³}\]

Key concepts

percentage calculation

Understanding percentage calculations is essential in various aspects of math and real-life applications. In our problem, we're given that one cup of Raisin Bran provides 60% of the U.S. Recommended Daily Allowance (RDA) of iron. To find the total RDA, we set up the equation illustrating that 60% of the RDA corresponds to 11 mg of iron. Remember, 60% as a decimal is 0.60.
This equation is written as:
\[ 0.60 \times \text{RDA} = 11 \text{ mg} \] By solving the equation, we isolate the RDA by dividing both sides by 0.60:
\[ \text{RDA} = \frac{11 \text{ mg}}{0.60} \approx 18.33 \text{ mg} \] Hence, our total U.S. RDA for iron is approximately 18.33 mg.

unit conversion

Unit conversion is an integral process in many scientific calculations. In this exercise, we need to convert milligrams (mg) to grams (g) because the density of iron is given in grams per cubic centimeter (g/cm³).
It's essential to know that 1 gram equals 1,000 milligrams. To convert 11 mg to grams, we use the conversion factor:
- \( 1 \text{ mg} = 0.001 \text{ g} \) So,
\[ 11 \text{ mg} = 11 \times 0.001 \text{ g} = 0.011 \text{ g} \] By converting to grams, we can now easily plug this value into further calculations involving density, which makes it all quite straightforward.

density formula

Density is a fundamental property that relates the mass of a substance to its volume, described by the formula:
\[ \text{density} = \frac{\text{mass}}{\text{volume}} \] We rearrange this formula to solve for volume:
\[ \text{volume} = \frac{\text{mass}}{\text{density}} \] In our example, iron's density is 7.874 g/cm³, and our converted mass is 0.011 g. Substituting these values into the rearranged equation gives:
\[ \text{volume} = \frac{0.011 \text{ g}}{7.874 \text{ g/cm}³} \approx 0.0014 \text{ cm}³ \] This result lets us understand how much physical space the pure extracted iron occupies in the cereal.

iron volume determination

After using unit conversion and the density formula, we can determine the volume of iron present in one cup of cereal. First, recall our calculated values: the mass of iron from one cup of Raisin Bran is 0.011 g, and its density is 7.874 g/cm³.
By substituting these values into the formula for volume, we get:
\[ \text{volume} = \frac{0.011 \text{ g}}{7.874 \text{ g/cm}³} \approx 0.0014 \text{ cm}³ \] This calculation tells us that the pure iron extracted from one cup of Raisin Bran will occupy a volume of approximately 0.0014 cubic centimeters. This small volume demonstrates how dense iron is.