An oil refinery is located 1 km north of the north bank of a straight river that is 3 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 8 km east of the refinery. The cost of laying pipe is $500,000/km over land to a point P on the north bank and $1,000,000/km under the river to the tanks. To minimize the cost of the pipeline, how far downriver from the refinery should the point P be located?
Draw a diagram. Let x be the distance east to point P
on land:
d^2 = x^2 + 1^2
underwater
d^2 = 3^2 + (8-x)^2
cost is thus (in half-millions)
c = sqrt(x^2 + 1) + 2sqrt(9 + (8-x)^2)
dc/dx = x/sqrt(x^2+1) - 2(8-x)/sqrt(9+(8-x)^2)
to find where dc/dx = 0 we put all over a common denominator, and set the numerator to zero:
xsqrt(9+(8-x)^2) - 2(8-x)sqrt(x^2+1) = 0
x^2(9 + (8-x)^2) = 4(8-x)^2(x^2+1)
This is satisfied when x is about 6.3
So, P should be 6.3 km downriver from the refinery. Cost will be about $M26.55
Better check my math for the details.
To minimize the cost of the pipeline, we need to determine the optimal position of point P on the north bank. This point will determine the distance downstream from the refinery where the pipeline should cross the river.
Let's break down the problem step by step:
1. Draw a diagram: Draw a diagram representing the problem. Label the locations of the refinery, point P on the north bank, and the storage tanks on the south bank of the river. Use appropriate scales and distances.
2. Determine the distance from the refinery to point P: Let's call this distance "x". Since the refinery is located 1 km north of the north bank of the river, the horizontal distance from the refinery to P is equal to x.
3. Determine the distance from point P to the storage tanks: Since the storage tanks are located 8 km east of the refinery, the horizontal distance from P to the storage tanks is 8 km - x.
4. Determine the distance from point P to the south bank of the river: Since the river is 3 km wide, the distance from P to the south bank of the river is 3 km.
5. Calculate the cost of laying pipe over land: The cost of laying pipe over land from the refinery to point P is $500,000/km, and the distance is x km. Therefore, the cost of the overland pipe is 500,000 * x.
6. Calculate the cost of laying pipe under the river: The cost of laying pipe under the river from point P to the storage tanks is $1,000,000/km, and the distance is 3 km (the width of the river). Therefore, the cost of the underwater pipe is 1,000,000 * 3.
7. Calculate the total cost of the pipeline: The total cost is the sum of the cost of laying pipe over land and under the river. So, the total cost is 500,000 * x + 1,000,000 * 3.
8. Minimize the total cost: We want to minimize the total cost of the pipeline. To do this, we can take the derivative of the total cost with respect to x, set it equal to zero, and solve for x. This will give us the value of x that minimizes the cost.
9. Using calculus, find the optimal value of x: Differentiate the total cost expression with respect to x and set it equal to zero:
d/dx (500,000 * x + 1,000,000 * 3) = 0
Simplifying, we get:
500,000 = 0
Since this equation does not have a solution, it means that the total cost is not a function of x. In other words, the cost will be the same regardless of the position of point P on the north bank.
Therefore, to minimize the cost of the pipeline, it doesn't matter how far downriver from the refinery the point P is located. The cost will remain constant at $500,000/km over land and $1,000,000/km under the river, regardless of the position of point P.
To minimize the cost of the pipeline, we need to find the optimal distance for point P on the north bank.
Let's assume that the distance downriver from the refinery to point P is 'x' km. Therefore, the remaining distance across the river from point P to the storage tanks will be (8 - x) km.
The cost of laying the pipeline over land is $500,000/km, so the cost from the refinery to point P is 500,000 * x dollars.
The cost of laying the pipeline under the river is $1,000,000/km, so the cost from point P to the storage tanks is 1,000,000 * (8 - x) dollars.
The total cost of the pipeline is the sum of these two costs, which is:
Cost = 500,000x + 1,000,000(8 - x)
To minimize the cost, we can find the derivative of the cost function and set it equal to zero:
d(Cost)/dx = 500,000 - 1,000,000 = 0
Simplifying the equation, we get:
-500,000 = -1,000,000x
Dividing by -1,000,000, we have:
x = 0.5
This means that the optimal distance downriver from the refinery to point P is 0.5 km.
Therefore, to minimize the cost of the pipeline, point P should be located 0.5 km downriver from the refinery.