The human body contains about \(10^{14}\) cells, and the diameter of a typical cell is about \(10 \mu \mathrm{m}\). Like all ordinary matter, cells are made of atoms; a typical atomic diameter is \(0.1 \mathrm{nm}\). How does the number of atoms in a cell compare with the number of cells in the body? a. greater \(\quad\) b. smaller \(\quad\) c. about the same

Volume estimation lies at the crux of comparing different scales in physics, bridging the gap between the microscopic and macroscopic worlds. To understand how different entities, like atoms and cells, compare in size, it’s essential to estimate their volumes. In our context, we approximate a cell as a sphere, leveraging the formula for a sphere’s volume: \(V = \frac{4}{3} \pi r^3\). This formulation simplifies complex shapes into more manageable mathematical terms.

When carrying out volume estimations, it is important to recognize that the objects in question are often not perfect geometric shapes. In biology, cells can have various and irregular shapes. However, for the purposes of estimation, treating a cell as a sphere provides a useful approximation. This approach enables us to gain insight into the relative size of cells when compared to atom volumes, which are also estimated in a similar spherical approximation. Estimating the volume of these tiny structures allows us to quantify and compare their sizes despite the vast differences in scales.

Scientific notation is a method of expressing numbers that are too large or too small to be conveniently written in decimal form. It is often used in science and engineering to simplify calculations and to express numbers in a standardized form. Scientific notation has two parts: a coefficient and a power of ten. For example, the scientific notation for the number 5,000 is \(5 \times 10^3\), and for 0.0005 is \(5 \times 10^{-4}\).

In the context of our volume estimation problem, we deal with numbers like \(5 \times 10^{-6}\) meters for the cell radius and \(5 \times 10^{-11}\) meters for the atom radius. Without scientific notation, these numbers would be cumbersome to write out and to work with in calculations. It's also much easier to compare the relative sizes of objects when they are expressed in scientific notation; the exponent provides a quick comparison of magnitude, crucial in a problem that involves comparing the vastly different scales of cells and atoms.

Comparing atoms and cells involves understanding their respective scales and how these scales relate to each other. Atoms are the fundamental building blocks of matter, incredibly tiny compared to the more complex structures they compose, like cells. Cells, as the smallest unit of life, are many times larger than atoms, comprising countless atoms in their structure.

While the actual number of atoms in a cell can vary widely depending on the type of cell and the elements involved, for exercise purposes, we make an estimation. By calculating volumes utilizing the sphere approximation, we can infer that a single cell occupies immensely more space than an individual atom. This inference leads us to understand that there are exponentially more atoms in each cell than there are cells in the human body. Commonly, this kind of comparison yields a sense of the vast quantity of atoms involved in constructing even the smallest biological structures we can see with the naked eye or basic microscopes, underlining the fascinating complexity of the biological and physical worlds.