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UNO [17]
2 years ago
13

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 e

xpensive 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.
Mathematics
1 answer:
zhuklara [117]2 years ago
5 0

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}

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