Answer :
The gecko is 6 orders of magnitude longer than the dust mite, measuring approximately 1,000,000 times its length.
1. Convert Units:
- Dust mite length: [tex]\(10^{-3}\)[/tex] millimeters
- Convert dust mite length to centimeters:
[tex]\(1\) millimeter = \(0.1\)[/tex] centimeters
So, [tex]\(10^{-3}\) millimeters = \(10^{-5}\)[/tex] centimeters
2. Calculate the Difference:
- Gecko length: [tex]\(10\)[/tex] centimeters
- Difference in lengths:
[tex]\(10\) centimeters - \(10^{-5}\) centimeters[/tex]
3. Express in Scientific Notation:
- The difference is approximately [tex]\(10^1 - 10^{-5}\)[/tex]centimeters
- This simplifies to [tex]\(10^{1 + 5}\)[/tex] centimeters
- Resulting in approximately [tex]\(10^6\)[/tex] centimeters
4. Determine Orders of Magnitude:
- Since the gecko is [tex]\(10^6\)[/tex] times longer than the dust mite, it's [tex]\(6\)[/tex] orders of magnitude longer.
Final answer:
A gecko that is 10 centimeters long is 10,000 times, or 4 orders of magnitude, longer than a dust mite that measures 10−3 millimeters in length.
Explanation:
To calculate how many orders of magnitude the gecko's length is compared to that of the dust mite, we first need to convert both measurements to the same unit. The dust mite is 10−3 millimeters (mm), which is the same as 10−6 meters (m), and the gecko is 10 centimeters (cm), which is the same as 10−2 meters (m).
Next, we divide the length of the gecko by the length of the mite:
10−2 m / 10−6 m = 10⁴
This means the gecko is 10,000 times longer than the mite, or 4 orders of magnitude longer. Orders of magnitude are a way to express the relative scale of numbers: a difference of one order of magnitude means a factor of ten.
In everyday terms, if the dust mite were the size of a small speck of dust, the gecko would be comparatively the length of some commonly used small classroom objects.
