Find two positive numbers that satisfy the given requirements. (Enter your answers as a comma-separated list.)

The product is 182 and the sum is a minimum.

We are tasked with finding two positive numbers whose product is 182, and for which the sum is minimized.

Let the two numbers be \( x \) and \( y \). We know:
- The product \( x \times y = 182 \), or \( xy = 182 \).
- The sum \( x + y \) needs to be minimized.

### Step 1: Express \( y \) in terms of \( x \)
From the equation \( xy = 182 \), we can solve for \( y \):
\[
y = \frac{182}{x}
\]

### Step 2: Write the sum \( S \)
The sum \( S \) of the two numbers is:
\[
S = x + y = x + \frac{182}{x}
\]

### Step 3: Minimize the sum
To minimize \( S \), we can take its derivative with respect to \( x \) and set it equal to zero to find the critical points.

\[
\frac{dS}{dx} = 1 - \frac{182}{x^2}
\]

Setting the derivative equal to zero:
\[
1 - \frac{182}{x^2} = 0
\]
\[
\frac{182}{x^2} = 1
\]
\[
x^2 = 182
\]
\[
x = \sqrt{182}
\]
\[
x \approx 13.49
\]

### Step 4: Find \( y \)
Since \( xy = 182 \), substitute \( x \approx 13.49 \) into the equation for \( y \):
\[
y = \frac{182}{13.49} \approx 13.49
\]

### Step 5: Conclusion
The two numbers are approximately \( x \approx 13.49 \) and \( y \approx 13.49 \). Since the product is exactly 182, the two numbers are both \( \sqrt{182} \).

Thus, the two positive numbers are \( 13.49 \) and \( 13.49 \).

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the maximum of a sum instead of the minimum?
2. What happens if the product is a negative number?
3. How can we confirm that the sum is minimized using the second derivative test?
4. What if the problem asked for an integer solution?
5. How can we apply this method to find two numbers whose difference is minimized?

**Tip:** Minimizing sums and maximizing products often involve symmetric relationships, so checking for equal numbers is a good strategy when looking for minima or maxima.

Find two positive numbers that satisfy the given requirements. The product is 182 and the sum is a minimum.

Find two positive numbers whose product is 182 and whose sum is minimized. This optimization problem involves algebraic manipulation, differentiation, and the application of the first derivative test to find the minimum sum. Suitable for high school students in Grades 10-12.

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