A helicopter is flying along the curve `y=x^(2)+2`. A soldier is placed at the point (3, 2). Find the nearest distance between the solider and the helicopter.

Answer

To find the nearest distance between the soldier located at the point (3, 2) and the helicopter flying along the curve \( y = x^2 + 2 \), we can follow these steps:

### Step 1: Define the Points
Let the point on the curve where the helicopter is located be \( (h, h^2 + 2) \). Here, \( h \) represents the x-coordinate of the helicopter's position on the curve.

### Step 2: Write the Distance Formula
The distance \( D \) between the soldier at point \( (3, 2) \) and the helicopter at point \( (h, h^2 + 2) \) can be expressed using the distance formula:
\[ D = \sqrt{(h - 3)^2 + (h^2 + 2 - 2)^2} \]
This simplifies to:
\[ D = \sqrt{(h - 3)^2 + (h^2)^2} \]

### Step 3: Minimize the Distance
To minimize the distance, we can minimize \( D^2 \) instead, as the square root function is monotonically increasing. Thus, we define:
\[ D^2 = (h - 3)^2 + (h^2)^2 \]
Expanding this, we get:
\[ D^2 = (h - 3)^2 + h^4 = (h^2 - 6h + 9) + h^4 = h^4 + h^2 - 6h + 9 \]

### Step 4: Find the Derivative
To find the minimum, we take the derivative of \( D^2 \) with respect to \( h \):
\[ \frac{d(D^2)}{dh} = 4h^3 + 2h - 6 \]

### Step 5: Set the Derivative to Zero
Setting the derivative equal to zero to find critical points:
\[ 4h^3 + 2h - 6 = 0 \]
Dividing the entire equation by 2 gives:
\[ 2h^3 + h - 3 = 0 \]

### Step 6: Find the Roots
To solve \( 2h^3 + h - 3 = 0 \), we can use the Rational Root Theorem or synthetic division. Testing \( h = 1 \):
\[ 2(1)^3 + (1) - 3 = 2 + 1 - 3 = 0 \]
Thus, \( h = 1 \) is a root. We can factor the polynomial:
\[ 2h^3 + h - 3 = (h - 1)(2h^2 + 2h + 3) \]
The quadratic \( 2h^2 + 2h + 3 \) has no real roots (discriminant \( < 0 \)), so the only real solution is \( h = 1 \).

### Step 7: Calculate the Minimum Distance
Now, substituting \( h = 1 \) back into the distance formula:
\[ D^2 = (1 - 3)^2 + (1^2)^2 = (-2)^2 + (1)^2 = 4 + 1 = 5 \]
Thus, the minimum distance \( D \) is:
\[ D = \sqrt{5} \]

### Final Answer
The nearest distance between the soldier and the helicopter is \( \sqrt{5} \).

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