To verify the athlete's statement that the distance between Paris La Défense and Invalides is more than 15 km, we need to calculate the straight-line distance between these two points. We can do this using the Euclidean distance formula. Here's a step-by-step solution to find this distance and compare it with 15 km.
### Step-by-Step Solution:
1. Identify the coordinates:
Let's denote the coordinates for Paris La Défense and Invalides as follows:
- La Défense: [tex]\( (48.890, 2.238) \)[/tex]
- Invalides: [tex]\( (48.8566, 2.3126) \)[/tex]
2. Calculate the difference in coordinates:
We first find the difference in the latitude and longitude between the two locations:
- Difference in latitude ([tex]\( \Delta \text{lat} \)[/tex]):
[tex]\[
\Delta \text{lat} = 48.890 - 48.8566 = 0.0334
\][/tex]
- Difference in longitude ([tex]\( \Delta \text{long} \)[/tex]):
[tex]\[
\Delta \text{long} = 2.238 - 2.3126 = -0.0746
\][/tex]
3. Use the Euclidean distance formula:
The Euclidean distance formula in two dimensions is:
[tex]\[
\text{distance} = \sqrt{(\Delta \text{lat})^2 + (\Delta \text{long})^2}
\][/tex]
Substituting the values we have:
[tex]\[
\text{distance} = \sqrt{(0.0334)^2 + (-0.0746)^2}
\][/tex]
[tex]\[
\text{distance} = \sqrt{0.00111556 + 0.00556516}
\][/tex]
[tex]\[
\text{distance} = \sqrt{0.00668072}
\][/tex]
[tex]\[
\text{distance} \approx 0.0817 \text{ degrees}
\][/tex]
4. Convert the distance from degrees to kilometers:
A rough conversion factor from degrees to kilometers is approximately 111 km per degree (considering the average distance along a great circle, noting that a degree of latitude is about 111 km).
[tex]\[
\text{distance in km} = 0.0817 \times 111
\][/tex]
[tex]\[
\text{distance in km} \approx 9.0787 \text{ km}
\][/tex]
5. Compare the calculated distance with the athlete's claim:
The calculated straight-line distance between Paris La Défense and Invalides is approximately 9.0787 km.
Therefore, the athlete's statement that the distance is more than 15 km is false. The actual distance is significantly less than 15 km.
