please help word problem?

Question

On a site map for offshore driling the centers of two large oil discoveries are located at A(3,10) and B(11,6). Oil from these sites will be piped to a refinery to be located on the shoreline represeted by the x axis. Determine the total length of pipe required if the refinery is located so as to minimize the length.

Answer 1

the answer will be 11 units

Answer 2

I assume you want L2 distances. That is, you want pipe run in a straight line from A to the refinery and a straight line from B to the refinery.

In that case, you will need 8*sqrt(5) units of pipe, or 17.8885 units of pipe.

The refinery will be located at (8,0), or 8 units on the x-axis.


Here's the solution method:

Let's say the refinery is located at (x,0). In that case, the amount of pipe needed is (using the distance formula):

f(x) = sqrt( (3-x)^2 + 10^2 ) + sqrt( (11-x)^2 + 6^2 )

You want to take the derivative of this function with respect to the variable x. You will then set that derivative equal to zero and solve for the x that allows that to happen. The derivative is:

f'(x) = 0.5*sqrt( x^2 - 6x + 109 )*(2x - 6) + 0.5*( x^2 - 22x + 157 )*(2x - 22)

If you set that equal to zero, you can subtract one term from both sides of the equation and then square both sides of the result. Then cross multiply and "foil" out all terms.

x^4 - 28x^3 + 298x^2 - 1140x + 1413 = x^4 - 28x^3 + 362x^2 - 3124x + 13189

Subtract the left side from the right and get

64x^2 - 1984x + 11776 = 0

Divide everything by 64 and get:

x^2 - 31x + 184 = 0

Now use the quadratic formula:

x = 0.5*( 31 +/- sqrt( 31^2 - 4*184 ) )

That gives us:

x = 0.5*( 31 +/- 15 )

which means

x = 23 or x = 8

However,

f(23) = 16*sqrt(5) = 35.7771
f(8) = 8*sqrt(5) = 17.8885

Clearly, x=8 is a minimum and x=23 is a maximum.

So you want f(8) units of pipe, or 8*sqrt(5) = 17.8885 units of pipe.

Answer 3

This problem can be solved by the distance formula

d = √(x₂- x₁) + (y₂- y₁)

d = √(3 -11)² + (10 - 6)²

d = √(- 8)² + (4)²

d = √64 + 16

d = √80

d = 8.94427191

d = 8.94 Rounded to two decimal places