To solve the problem, we can use the coordinate system to represent the positions of Clarence's house (C), Ashland (A), and Oak Grove (O).
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Let's place Clarence's house at the origin of our coordinate system: \[ C(0, 0) \]
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Since Ashland is due east of Clarence's house and 7 miles away, we can place Ashland at: \[ A(7, 0) \]
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Given that Oak Grove is due north of Clarence's house and also related to Ashland, we first need to find the coordinates for Oak Grove. We know that Ashland is 9 miles from Oak Grove.
Let the coordinates of Oak Grove be \( O(x, y) \). The distance between A and O is then calculated as follows: \[ \sqrt{(x - 7)^2 + (y - 0)^2} = 9 \]
Squaring both sides gives: \[ (x - 7)^2 + y^2 = 81 \]
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Assuming Oak Grove is directly north of Clarence's house (which is due south), its x-coordinate must be the same as Clarence's, hence: \[ x = 0 \]
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Substituting \( x = 0 \) into the distance equation provides: \[ (0 - 7)^2 + y^2 = 81 \] \[ 49 + y^2 = 81 \] \[ y^2 = 81 - 49 \] \[ y^2 = 32 \] \[ y = \sqrt{32} = 4\sqrt{2} \approx 5.66 \quad (\text{Taking the positive root since Oak Grove is north}) \]
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Now we can calculate the distance from Clarence's house to Oak Grove (C to O): \[ \text{Distance} = \sqrt{(0 - 0)^2 + (0 - 4\sqrt{2})^2} = \sqrt{(4\sqrt{2})^2} = \sqrt{32} = 4\sqrt{2} \approx 5.66 \]
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Rounding to the nearest tenth, the distance from Oak Grove to Clarence's house is: \[ \text{Distance} \approx 5.7 \text{ miles} \]
To summarize, the straight line distance from Clarence's house to Oak Grove is approximately: \[ \boxed{5.7} \] miles.
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