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To find the point on the graph of \( f(x) = \sqrt{x - 7} \) that is closest to the point \( (10, 0) \), we can proceed with the following steps:

1. **Define the Distance Function:**  
   The distance \( d \) between a point on the curve \( (x, f(x)) \) and the point \( (10, 0) \) is given by:
   \[
   d = \sqrt{(x - 10)^2 + \left( \sqrt{x - 7} - 0 \right)^2} = \sqrt{(x - 10)^2 + (x - 7)}
   \]

2. **Simplify the Distance Function:**  
   To make it easier to work with, we'll minimize the square of the distance instead of \( d \) itself (since the square root is a monotonic function, this won't change the minimum point):
   \[
   d^2 = (x - 10)^2 + (x - 7)
   \]
   Expanding this expression:
   \[
   d^2 = (x - 10)^2 + x - 7 = x^2 - 20x + 100 + x - 7 = x^2 - 19x + 93
   \]

3. **Find the Minimum of \( d^2 \):**  
   To minimize \( d^2 \), we take the derivative with respect to \( x \) and set it to zero:
   \[
   \frac{d}{dx}(d^2) = 2x - 19 = 0
   \]
   Solving for \( x \):
   \[
   x = \frac{19}{2} = 9.5
   \]

4. **Find \( y \) Value on the Curve:**  
   Now, substitute \( x = 9.5 \) into \( f(x) \) to find \( y \):
   \[
   f(9.5) = \sqrt{9.5 - 7} = \sqrt{2.5}
   \]

So, the point on the graph closest to \( (10, 0) \) is:
\[
(x, y) = \left( 9.5, \sqrt{2.5} \right)
\]

Would you like a more detailed explanation, or have any questions?

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### Related Questions:

1. How do you find the minimum distance from a point to a line instead of a curve?
2. Why does minimizing \( d^2 \) yield the same result as minimizing \( d \)?
3. Can the method used here be applied to other functions, like polynomial functions?
4. How would the process change if the point were further away, say at \( (20, 0) \)?
5. How can we use calculus to verify that our minimum distance is indeed the closest point?

### Tip:
In optimization problems involving distance, minimizing the square of the distance often simplifies calculations by removing the square root, while still preserving the location of the minimum.

Find the point on the graph of the function that is closest to the given point: f(x) = √(x - 7), (10, 0).

Learn how to find the point on the graph of the function f(x) = √(x - 7) that is closest to the point (10, 0) using distance minimization techniques and calculus. This problem involves concepts like distance calculation, derivatives, and optimization. Suitable for students in Grades 11-12.

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