To determine the total distance the bird flew, we need to find the distances of each leg of the journey: from the courthouse to the library, from the library to the swimming pool, and from the swimming pool back to the courthouse.
Step 1: Setup the Coordinates
Let’s assume the following coordinates based on the descriptions:
- Courthouse: \((0, 0)\)
- Library: \((0, -6.9)\) (6.9 miles south of the courthouse)
- Swimming Pool: \((-3.3, -6.9)\) (3.3 miles west of the library)
Step 2: Calculate Distances
-
Distance from Courthouse to Library:
- This distance is simply the southward distance:
\[ D_{CL} = 6.9 \text{ miles} \]
- This distance is simply the southward distance:
-
Distance from Library to Swimming Pool:
- The change in x-coordinates: \(-3.3 - 0 = -3.3\)
- The change in y-coordinates: \(-6.9 - (-6.9) = 0\)
- This forms a horizontal movement westward, simply the distance to the west:
\[ D_{LS} = 3.3 \text{ miles} \]
-
Distance from Swimming Pool to Courthouse:
- The change in x-coordinates: \(-3.3 - 0 = -3.3\)
- The change in y-coordinates: \(-6.9 - 0 = -6.9\)
- Using the Pythagorean theorem to find the distance:
\[ D_{SC} = \sqrt{(-3.3 - 0)^2 + (-6.9 - 0)^2} \] - Calculating:
\[ D_{SC} = \sqrt{(-3.3)^2 + (-6.9)^2} = \sqrt{10.89 + 47.61} = \sqrt{58.5} \] - Approximating \(\sqrt{58.5}\):
\[ \sqrt{58.5} \approx 7.65 \text{ miles} \]
Step 3: Total Distance
Now, we can find the total distance flown by adding up the individual distances: \[ \text{Total Distance} = D_{CL} + D_{LS} + D_{SC} \] \[ \text{Total Distance} = 6.9 + 3.3 + 7.65 \approx 17.85 \text{ miles} \]
Final Answer
Rounding to the nearest tenth, the total distance flown is: \[ \boxed{17.9 \text{ miles}} \]
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