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Chapter 4: Problem 4

An ant is roughly \(10^{-3}\) meter in length and the average human roughly one meter. How many times longer is a human than an ant?

Short Answer

Expert verified
A human is 1000 times longer than an ant.

Step by step solution

01

- Understand the Units

First, note the lengths given: the ant is approximately \(10^{-3}\) meters and the human is approximately 1 meter.
02

- Set Up the Ratio

To find how many times longer a human is compared to an ant, set up the ratio of the human's length to the ant's length, which is \( \frac{1}{10^{-3}} \).
03

- Calculate the Ratio

Calculate the ratio by performing the division: \( \frac{1}{10^{-3}} = 10^3 \).
04

- Interpret the Result

The result \( 10^3 \) means that a human is 1000 times longer than an ant.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

metric units
In this exercise, we deal with measurements in meters. Meters are the base unit of length in the International System of Units (SI).
When we say an ant is approximately \(10^{-3}\) meters, this means the ant's length is 0.001 meters, or one millimeter.
On the other hand, the average human is roughly 1 meter tall. It is crucial to understand metric units to easily convert between different sizes and scales.
If we were dealing with lengths in other units like centimeters, we would say:\r
    \r
  • The ant is about 0.1 centimeters (cm)
    • \r
  • The human is 100 centimeters (cm)
    • \r
\r
Knowing how to convert and understand these units helps greatly in solving and interpreting ratio problems.
division
To find how many times longer a human is than an ant, we use division.
We set up the ratio of the human鈥檚 length to the ant鈥檚 length: \( \frac{1}{10^{-3}} \).
Division is simply splitting a number into equal parts or groups.
In this case, we are dividing the human鈥檚 length by the ant鈥檚 length.

Remember these key points about division: \r
    \r
  • It is the opposite operation of multiplication.
    • \r
  • When you divide by a number less than 1, the result is greater than the initial number.
    • \r
  • Division of fractions and powers often requires multiplication of their reciprocals.
    • \r
\r
In simpler terms, \( \frac{1}{10^{-3}} \) divides 1 by a small fraction, hence expands the value significantly.
exponents
Exponents, or powers, are a shorthand way of showing that a number is to be multiplied by itself a certain number of times.
In the equation \(10^{-3}\), the exponent -3 shows that 10 is divided by itself three times (i.e., \(10^{-3} = \frac{1}{10^3} = 0.001\)).
Exponents are particularly useful in comparing vastly different scales, like comparing the size of an ant to a human.
We use the properties of exponents to simplify division calculations.\r
\rKey points: \r
    \r
  • Positive exponents indicate multiplication (e.g., \(10^3 = 10 \times 10 \times 10 = 1000\)).
    • \r
  • Negative exponents indicate division (e.g., \(10^{-3} = \frac{1}{10^3} = 0.001\)).
    • \r
  • Division of exponents involves subtracting the exponents if they have the same base.
    For example, \( \frac{10^1}{10^{-3}} = 10^{1 - (-3)} = 10^4 \).
    • \r
\r
Using these rules, dividing 1 meter by \(10^{-3}\) meters simplifies to \(10^3\), revealing that a human is 1000 times longer than an ant.

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