Calculating Lung Volume in Humans. In humans, oxygen and...

Calculating Lung Volume in Humans. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $480 \times 10^{6}$ and that the average volume of a single alveolus is $4.2 \times 10^{6} \mu \mathrm{m}^{3} .$ (The volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ and the area of a sphere is $\left.A=4 \pi r^{2} .\right)$ If we assume that alveoli are spherical, what is the diameter of a typical alveolus? (a) $0.20 \mathrm{~mm} ;$ (b) $2 \mathrm{~mm} ;$ (c) $20 \mathrm{~mm} ;$ (d) $200 \mathrm{~mm}$.

Calculating Lung Volume in Humans. In humans, oxygen and carbon dioxide are exchanged in the blood within many small sacs called alveoli in the lungs. Alveoli provide a large surface area for gas exchange. Recent careful measurements show that the total number of alveoli in a typical pair of lungs is about $480 \times 10^{6}$ and that the average volume of a single alveolus is $4.2 \times 10^{6} \mu \mathrm{m}^{3} .$ (The volume of a sphere is $V=\frac{4}{3} \pi r^{3},$ and the area of a sphere is $\left.A=4 \pi r^{2} .\right)$
If we assume that alveoli are spherical, what is the diameter of a
typical alveolus? (a) $0.20 \mathrm{~mm} ;$ (b) $2 \mathrm{~mm} ;$ (c) $20 \mathrm{~mm} ;$ (d) $200 \mathrm{~mm}$.

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Key Concepts

Spherical Geometry

Spherical geometry deals with the properties and measurements of spheres. In many scientific contexts, objects are approximated as spheres to simplify complex shapes. This abstraction is important for calculating volumes and surface areas, which are crucial in fields ranging from biology to physics.

Volume of a Sphere

The formula for the volume of a sphere, V = (4/3)?r³, is fundamental in relating the size of an object to its spatial capacity. By knowing the volume, one can deduce the radius or diameter of a sphere, a technique widely applicable in both theoretical and applied sciences.

Algebraic Manipulation

Solving for an unknown variable, such as the radius or diameter from the sphere's volume, requires rearranging equations and applying operations such as taking cube roots. This skill of algebraic manipulation is essential in many areas of mathematics and science for deriving quantities from known relationships.

Unit Conversion and Dimensional Analysis

Unit conversion ensures that all measurements are in consistent units, which is critical for accurate calculations. Dimensional analysis checks that equations make sense in terms of their physical units, a practice that is indispensable across scientific disciplines when translating between scales or measurement systems.

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