Problem 14 - Astronomy: At Play in the Cosmos

Question

Comparing the size of a human$$\left(L_{\mathrm{HB}}=10^{\circ} \text { meters }\right)$$to the size of an \right. atom \(\left(L=10^{-10} \text {meters }\right)\) a. the human is 9 times bigger. b. the human is 10 times bigger c. the atom is 9 orders of magnitude smaller. d. the human is a billion times bigger. e. the atom is 10 powers of 10 smaller.

Step-by-Step Solution

Answer

Option e is correct: the atom is 10 powers of 10 smaller.

Step 1: Define the Concept of Orders of Magnitude

The term 'orders of magnitude' refers to the logarithm base 10 difference of two numbers. It describes how many times you would multiply the smaller number by 10 to equal the larger number.

Step 2: Understand the Problem Values

We are given two sizes: a human size, \(L_{\mathrm{HB}} = 10^0\) meters, and an atom size, \(L = 10^{-10}\) meters.

Step 3: Calculate the Difference in Orders of Magnitude

To determine how many orders of magnitude the human size is larger than the atom size, subtract the exponent of the atom size from the exponent of the human size: \(0 - (-10) = 10\).

Step 4: Interpret the Difference

A difference of 10 in orders of magnitude means that the human size is equivalent to \(10^{10}\) times the size of an atom.

Step 5: Determine the Correct Answer

From the calculation, we find that the human size is 10 orders of magnitude larger than the atom size, which corresponds to \(10^{10}\). This means the atom is 10 powers of 10 smaller.

Key Concepts

Understanding Size Comparison

Size comparison helps us understand how a measurement or size differs relative to another. At its core, it involves determining which of two items is larger or smaller, and by what factor. For instance, when comparing the size of a human to an atom, we look at how much larger a human is compared to the very tiny size of an atom.
In the context of the original exercise, we use a method called 'orders of magnitude' to help us make sense of this size difference. To put it simply, it tells us how many times one object is bigger or smaller when expressed as a power of 10.
This method simplifies huge or tiny numbers into more understandable terms.

• A human is represented as approximately 1 meter in size, or mathematically, \(10^0\) meters.
• An atom, on the other hand, is a mere \(10^{-10}\) meters in size.

Through size comparison, we can see just how vast the difference is!

Logarithm Base 10

The concept of a logarithm base 10 is like using shortcuts to handle big numbers.
Instead of multiplying so many times, we use the base 10 logarithm to work out how many times we multiply the number 10 to get another number.
This is super handy in scientific notation where expressions are written as powers of 10.
In the original exercise, you encountered:

• The human size as \(10^0\) meters.
• The atom's size as \(10^{-10}\) meters.

When you note the difference using base 10 logarithm:

• You subtract the exponent of the atom from that of the human, resulting in \(0 - (-10)\).
• This simplifies to 10, which represents how many times you need to multiply the atom’s size by 10 to bridge the gap to human size.

Using base 10, complexities become easier to grasp, especially in comparing extremely different sizes.

Powers of 10

Powers of 10 are like magic tools that let us describe really big or really tiny numbers easily.Whenever you see something like \(10^n\), it means ‘10 multiplied by itself n times’.
Say, for \(10^2\), you multiply 10 two times: 10 x 10 = 100.In comparing sizes, it becomes essential because it gives a clear understanding of magnitude.
In the exercise, both a human and an atom are expressed as powers of 10.

• Human: \(10^0\)
• Atom: \(10^{-10}\)

The difference of 10 in exponents means there is a factor of \(10^{10}\) and thus, the human is astronomically larger when compared in terms of powers of 10.
This powers of 10 scale avoids getting lost in zeros, providing a neat representation of size differences.

Human Size vs Atom Size

Let’s dive into the mind-boggling difference between human size and atom size.Humans measure around 1 meter (3.28 feet) tall on average.An individual atom is unbelievably tiny, around \(10^{-10}\) meters in size!
To make sense of this huge gap, scientists use orders of magnitude, showing that we're dealing with differences of \(10^{10}\).That means, the size of a human is \(10^{10}\) times larger than that of an atom.
If you imagined stacking \(10^{-10} \) sized atoms to equal the height of a person, you would need one billion atoms!

• This ordered magnitude approach helps simplify comparisons in sciences and mathematics.
• The human vs atom size comparison not only shows biological dominance but also scientific marvel on how the universe fits vastness and tiny particles in harmony.

Understanding these comparisons gives insight into the vast scales present in our universe.