# Math

posted by .

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?

• Math -

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.

## Similar Questions

1. ### Calculus

Can anyone help me with these two optimization problems?
2. ### Calc

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 3 km east of the refinery. The cost of laying …
3. ### MATH

It is a well-known fact that homing pigeons fly faster over land than over water. Assume that they fly at 10 metres per second over land, and only 8 metres per second over water. A pigeon is located at the edge of a straight river …

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to storage tanks located on the south bank of the river 6 km east of the refinery. The cost of laying …
5. ### Physics

A 400.0-m-wide river flows from west to east at 30.0 m/min. Your boat moves at 100.0 m/min relative to the water no matter which direction you point it. To cross this river, you start from a dock at point A on the south bank. There …
6. ### physics

A 410-m-wide river has a uniform flow speed of 2.0 m/s through a jungle and toward the east. An explorer wishes to leave a small clearing on the south bank and cross the river in a powerboat that moves at a constant speed of 7.9 m/s …
7. ### Optimization

A hunter is at a point on a river bank. he wants to get his cabin, located 19 miles north and 8 miles west. He can travel 5 mph on the river bank and 2 mph on the rough rocky ground.how far upriver should he go in order to reach the …
8. ### Math

An oil company owns two refineries. The daily production limits and operating costs for each refinery are given in the table below. An order is received for 1540 barrels of high-grade oil, 1650 barrels of medium-grade oil, and 2860 …
9. ### Physics

The captain of a boat wants to travel directly across a river that flows due east with a speed of 1.09 m/s. He starts from the south bank of the river and wants to reach the north bank by travelling straight across the river. The boat …
10. ### Need Help Physics

The captain of a boat wants to travel directly across a river that flows due east with a speed of 1.09 m/s. He starts from the south bank of the river and wants to reach the north bank by travelling straight across the river. The boat …

More Similar Questions