To find out how much distance Keith would have saved if he had ridden directly to the grocery store, we can apply the Pythagorean theorem. Let's denote the distance Keith rode down the first road as \( x \) miles.
After riding \( x \) miles, he took a 90-degree turn and rode 2 miles down another road. The two roads form a right triangle where \( x \) and 2 are the two legs of the triangle, and the direct distance to the grocery store is the hypotenuse \( d \).
According to the Pythagorean theorem:
\[ d = \sqrt{x^2 + 2^2} \]
Now we need to calculate the total distance Keith rode, which is \( x + 2 \).
The distance saved by taking the direct route is:
\[ \text{Distance saved} = (x + 2) - d \]
Substituting for \( d \):
\[ \text{Distance saved} = (x + 2) - \sqrt{x^2 + 4} \]
Next, we will find the expression for the distance saved. To find a numerical measure, we just need to establish \( x \). However, since the value of \( x \) is not provided, we cannot compute the exact saved distance without that information. Nevertheless, we can rewrite the formula for any given \( x \).
Let's assume \( x \) is a specific value. For example, if we assume \( x = 3 \):
Then, we will have:
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The total distance Keith actually rode: \[ 3 + 2 = 5 \text{ miles} \]
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The direct distance using \( x = 3 \): \[ d = \sqrt{3^2 + 2^2} = \sqrt{9 + 4} = \sqrt{13} \approx 3.60555 \text{ miles} \text{ (rounded to 5 decimal places)} \]
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The distance saved: \[ \text{Distance saved} = 5 - 3.60555 \approx 1.39445 \text{ miles} \] Rounding to the nearest tenth, that would be approximately: \[ 1.4 \text{ miles} \]
In conclusion, given a specific value for \( x \), we can compute the corresponding distance saved. If you have a specific value for \( x \), please provide it for exact results. Otherwise, the process remains valid for any \( x \).
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