Answer :
To find out how many orders of magnitude longer the gecko is compared to the dust mite, we first need to compare their lengths in a common unit and then calculate the difference in their orders of magnitude.
Here's a step-by-step breakdown:
1. Convert Lengths to Meters:
- The dust mite measures [tex]\(10^{-3}\)[/tex] millimeters in length. To convert this to meters, divide by 1,000 (since there are 1,000 millimeters in a meter):
[tex]\[
\text{Dust mite length in meters} = \frac{10^{-3}}{1000} = 10^{-6} \text{ meters}
\][/tex]
- The gecko measures 10 centimeters in length. To convert centimeters to meters, divide by 100 (since there are 100 centimeters in a meter):
[tex]\[
\text{Gecko length in meters} = \frac{10}{100} = 0.1 \text{ meters}
\][/tex]
2. Calculate the Orders of Magnitude:
- Orders of magnitude compare the scale of two quantities. This can be done by dividing one length by the other and taking the base 10 logarithm:
[tex]\[
\text{Orders of magnitude} = \log_{10} \left(\frac{\text{Gecko length in meters}}{\text{Dust mite length in meters}}\right)
\][/tex]
- Substitute the lengths in meters:
[tex]\[
= \log_{10} \left(\frac{0.1}{10^{-6}}\right)
\][/tex]
When you perform this calculation, the result is:
[tex]\[
5.0
\][/tex]
This result means that the gecko is 5 orders of magnitude longer than the dust mite. An order of magnitude represents a factor of ten, so the gecko is [tex]\(10^5\)[/tex] times longer than the mite.
Here's a step-by-step breakdown:
1. Convert Lengths to Meters:
- The dust mite measures [tex]\(10^{-3}\)[/tex] millimeters in length. To convert this to meters, divide by 1,000 (since there are 1,000 millimeters in a meter):
[tex]\[
\text{Dust mite length in meters} = \frac{10^{-3}}{1000} = 10^{-6} \text{ meters}
\][/tex]
- The gecko measures 10 centimeters in length. To convert centimeters to meters, divide by 100 (since there are 100 centimeters in a meter):
[tex]\[
\text{Gecko length in meters} = \frac{10}{100} = 0.1 \text{ meters}
\][/tex]
2. Calculate the Orders of Magnitude:
- Orders of magnitude compare the scale of two quantities. This can be done by dividing one length by the other and taking the base 10 logarithm:
[tex]\[
\text{Orders of magnitude} = \log_{10} \left(\frac{\text{Gecko length in meters}}{\text{Dust mite length in meters}}\right)
\][/tex]
- Substitute the lengths in meters:
[tex]\[
= \log_{10} \left(\frac{0.1}{10^{-6}}\right)
\][/tex]
When you perform this calculation, the result is:
[tex]\[
5.0
\][/tex]
This result means that the gecko is 5 orders of magnitude longer than the dust mite. An order of magnitude represents a factor of ten, so the gecko is [tex]\(10^5\)[/tex] times longer than the mite.
