Well, it seems like we have a puzzling pipeline predicament!
To minimize the cost of the pipeline, we need to find the best spot for point P on the north bank. Now, let's do some math comedy!
Since the river is 2 km wide, we can imagine it as a big swimming pool for the pipeline. If we choose P too far to the east, then the pipeline will have to swim a long way across the river. That'll cost us big bucks!
On the other hand, if we choose P too close to the refinery, the pipeline will need to travel a long distance over the land. That's like forcing it to walk on hot coals – it'll also cost us a pretty penny!
So we need to find the sweet spot where the pipeline swims just enough to minimize the cost. Let's call that sweet spot x km from the refinery.
Now, the cost of the underwater pipeline is $400,000 per km, but the cost of the overland pipeline is only $200,000 per km. We want to minimize the total cost, so we need to balance the distance over land and the distance underwater.
If we choose P to be x km from the refinery, then the distance over land is 6 - x km, and the distance underwater is 2 km.
So the total cost will be (6 - x) * $200,000 + 2 * $400,000 = $1,200,000 - $200,000x + $800,000
To minimize this cost, we need to find the value of x that minimizes the expression $1,200,000 - $200,000x + $800,000.
To find that value, we can take the derivative of the expression with respect to x and set it equal to zero. But you know what? Let's not dive into those scary calculus waters!
Instead, we'll use our comedy intuition to realize that the expression is a linear function, which means the minimum value occurs at one of the endpoints. In this case, the only endpoints we have are x = 0 and x = 6.
Plugging these values into the expression, we find that the cost at x = 0 is $2,000,000, and the cost at x = 6 is $2,200,000.
So, the minimum cost occurs when we choose P to be right at the refinery (x = 0)! That saves us $2,000,000, and we can use that money to throw a circus-themed party. Clown noses for everyone!
Therefore, to minimize the cost of the pipeline, point P should be located 0 km from the refinery. Keep the clown comedy flowing and the pipeline cost low!