Question

An offshore oil well is 1 mile off the coast. The oil refinery is 4 miles down the coast. Laying pipe in the ocean is twice as expensive as laying it on land. Find x, and consider how finding x helps determine the most economical path for the pipe from the well to the oil refinery.

Answer 1

Answer:

$x = \frac{\sqrt 3}{ 3}$

Step-by-step explanation:

Given

See attachment for complete question

First, we calculate the water pipe length (AC):

$AC^2 = AB^2 + BC^2$

$AC^2 = 1^2 + x^2$

$AC^2 = 1 + x^2$

$AC = \sqrt{1 + x^2$

The cost of laying across water is twice (2 times) laying on land.

So, the total cost (C) is:

$C = 2 * \sqrt{1 + x^2} + 1 * (4 - x)$

$C = 2 \sqrt{1 + x^2} + (4 - x)$

Differentiate

$C' = \frac{2x}{\sqrt{1 + x^2}} - 1$

To find the most economical cost, we simply minimize C' by equating C' to 0

$C' = 0$

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

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

Cross multiply

$2x= \sqrt{1 + x^2}$

Take square of both sides

$4x^2 = 1+x^2$

Collect like terms

$4x^2 -x^2= 1$

$3x^2= 1$

Solve for $x^2$

$x^2 = \frac{1}{3}$

Solve for x

$x = \sqrt{\frac{1}{3}}$

$x = \frac{1}{\sqrt 3}$

Rationalize:

$x = \frac{1}{\sqrt 3} * \frac{\sqrt 3}{\sqrt 3}$

$x = \frac{\sqrt 3}{ 3}$