Answer:
[tex]x = \frac{\sqrt 3}{ 3}[/tex]
Step-by-step explanation:
Given
See attachment for complete question
First, we calculate the water pipe length (AC):
[tex]AC^2 = AB^2 + BC^2[/tex]
[tex]AC^2 = 1^2 + x^2[/tex]
[tex]AC^2 = 1 + x^2[/tex]
[tex]AC = \sqrt{1 + x^2[/tex]
The cost of laying across water is twice (2 times) laying on land.
So, the total cost (C) is:
[tex]C = 2 * \sqrt{1 + x^2} + 1 * (4 - x)[/tex]
[tex]C = 2 \sqrt{1 + x^2} + (4 - x)[/tex]
Differentiate
[tex]C' = \frac{2x}{\sqrt{1 + x^2}} - 1[/tex]
To find the most economical cost, we simply minimize C' by equating C' to 0
[tex]C' = 0[/tex]
[tex]\frac{2x}{ \sqrt{1 + x^2}}-1 = 0\\[/tex]
[tex]\frac{2x}{ \sqrt{1 + x^2}}=1[/tex]
Cross multiply
[tex]2x= \sqrt{1 + x^2}[/tex]
Take square of both sides
[tex]4x^2 = 1+x^2[/tex]
Collect like terms
[tex]4x^2 -x^2= 1[/tex]
[tex]3x^2= 1[/tex]
Solve for [tex]x^2[/tex]
[tex]x^2 = \frac{1}{3}[/tex]
Solve for x
[tex]x = \sqrt{\frac{1}{3}}[/tex]
[tex]x = \frac{1}{\sqrt 3}[/tex]
Rationalize:
[tex]x = \frac{1}{\sqrt 3} * \frac{\sqrt 3}{\sqrt 3}[/tex]
[tex]x = \frac{\sqrt 3}{ 3}[/tex]
