This problem can be approached using geometric optimization principles, specifically the Fermat Point concept, and calculus to minimize the total pipeline length.

Problem Setup:

1. Plants and Lake Layout:

   • The two manufacturing plants are 12 km apart.
   • The pipeline from the lake meets at a point $P$, which minimizes the total length of the pipes connecting the plants to $P$ and $P$ to the lake.

2. Optimization Variable: Let the distance from the point $P$ to the lake be $x$, where $x$ lies along the perpendicular line connecting the midpoint of the plants and the lake.

Total Length of the Pipeline ($L$):

The total length of the pipeline consists of two segments from the plants to $P$, plus the segment from $P$ to the lake.

The distance from $P$ to the plants forms a right triangle for each segment, using the Pythagorean Theorem. Let the height from $P$ to the line joining the two plants be $h$.

$h = \sqrt{10^2 - x^2}$

Thus, the total length of the pipeline, $L(x)$, is:

$L(x) = 2 \sqrt{\left(\frac{12}{2}\right)^2 + h^2} + x$

Substituting $h = \sqrt{10^2 - x^2}$:

$L(x) = 2 \sqrt{\left(\frac{12}{2}\right)^2 + (10^2 - x^2)} + x$

$L(x) = 2 \sqrt{6^2 + (100 - x^2)} + x$

$L(x) = 2 \sqrt{36 + 100 - x^2} + x$

$L(x) = 2 \sqrt{136 - x^2} + x$

Minimizing $L(x)$:

To minimize $L(x)$, take its derivative and set it equal to zero.

1. Differentiate $L(x)$:

[ L'(x) = \frac{d}{dx}\left[ 2 \sqrt{136 - x^2} \right] + \frac{d}{dx}[x] ]

$L'(x) = 2 \cdot \frac{-x}{\sqrt{136 - x^2}} + 1$

2. Set $L'(x) = 0$:

$\frac{-2x}{\sqrt{136 - x^2}} + 1 = 0$

$\frac{-2x}{\sqrt{136 - x^2}} = -1$

$\frac{2x}{\sqrt{136 - x^2}} = 1$

$2x = \sqrt{136 - x^2}$

3. Square both sides:

$(2x)^2 = 136 - x^2$

$4x^2 = 136 - x^2$

$5x^2 = 136$

$x^2 = \frac{136}{5}$

$x = \sqrt{\frac{136}{5}}$

$x \approx 5.21 \, \text{km}$

Verifying Minimum:

To confirm this is a minimum, check the second derivative $L''(x)$:

$L''(x) = \frac{d}{dx}\left[ \frac{-2x}{\sqrt{136 - x^2}} \right]$

Simplify and evaluate $L''(x) > 0$, ensuring concavity upwards.

Final Result:

The distance $x$ from the lake to $P$ that minimizes the pipeline is approximately 5.21 km. This ensures the total pipeline length is minimized.

Would you like me to expand on the verification or calculate the exact pipeline length for this optimized configuration?

Related Questions:

1. How would the result change if the lake was closer or farther from the plants?
2. Can this method be extended to more than two plants? If yes, how?
3. What are the real-life challenges of constructing an underground pipeline using such optimization?
4. What is the practical significance of minimizing the pipeline in terms of costs and resources?
5. How can we adapt the model if the lake is not directly perpendicular to the plants?

Tip:

Always double-check units and assumptions (e.g., distances and perpendicularity) when solving real-world optimization problems!