Answer :
Sure! Let's break down the solution to find out how many orders of magnitude longer the gecko is compared to the dust mite.
1. Understand the Measurements:
- The dust mite measures [tex]\(10^{-3}\)[/tex] millimeters in length.
- The gecko measures 10 centimeters in length.
2. Convert Measurements to the Same Unit:
- Since the dust mite's length is in millimeters, we'll convert the gecko's length from centimeters to millimeters for a direct comparison.
- Recall that [tex]\(1\)[/tex] centimeter equals [tex]\(10\)[/tex] millimeters. So, to convert the gecko's length from centimeters to millimeters, we multiply by [tex]\(10\)[/tex]:
[tex]\[
\text{Gecko length in mm} = 10 \, \text{cm} \times 10 \, \left(\frac{\text{mm}}{\text{cm}}\right) = 100 \, \text{mm}
\][/tex]
3. Calculate the Orders of Magnitude:
- The order of magnitude difference is found by dividing the gecko's length by the dust mite's length and then taking the base 10 logarithm:
[tex]\[
\frac{\text{Gecko's length in mm}}{\text{Mite's length in mm}} = \frac{100}{0.001} = 100,000
\][/tex]
- The number of orders of magnitude is the power to which 10 must be raised to get 100,000. This is found using the base 10 logarithm:
[tex]\[
\log_{10}(100,000) = 5
\][/tex]
Therefore, the gecko is 5 orders of magnitude longer than the dust mite.
1. Understand the Measurements:
- The dust mite measures [tex]\(10^{-3}\)[/tex] millimeters in length.
- The gecko measures 10 centimeters in length.
2. Convert Measurements to the Same Unit:
- Since the dust mite's length is in millimeters, we'll convert the gecko's length from centimeters to millimeters for a direct comparison.
- Recall that [tex]\(1\)[/tex] centimeter equals [tex]\(10\)[/tex] millimeters. So, to convert the gecko's length from centimeters to millimeters, we multiply by [tex]\(10\)[/tex]:
[tex]\[
\text{Gecko length in mm} = 10 \, \text{cm} \times 10 \, \left(\frac{\text{mm}}{\text{cm}}\right) = 100 \, \text{mm}
\][/tex]
3. Calculate the Orders of Magnitude:
- The order of magnitude difference is found by dividing the gecko's length by the dust mite's length and then taking the base 10 logarithm:
[tex]\[
\frac{\text{Gecko's length in mm}}{\text{Mite's length in mm}} = \frac{100}{0.001} = 100,000
\][/tex]
- The number of orders of magnitude is the power to which 10 must be raised to get 100,000. This is found using the base 10 logarithm:
[tex]\[
\log_{10}(100,000) = 5
\][/tex]
Therefore, the gecko is 5 orders of magnitude longer than the dust mite.
