Respuesta :
Answer:
Length of pipe to Chesterville is 8.376 miles and
Length of pipe to Denton is 5.46 miles
Step-by-step explanation:
Here we have
The distance of Chesterville from the river is 3 miles, while the distance of Denton from the river is 5 miles
The bank of the river is 10 miles long
Therefore, we have
If x is the distance from the point directly opposite to Chesterville to the location of the water works, the equation is;
Cost to Chesterville = [tex]3000\times \sqrt{x^2 + 3^2}[/tex]
Cost to Denton = [tex]7000\times \sqrt{(10-x)^2 + 5^2}[/tex]
Total cost is then;
[tex]7000\times \sqrt{(10-x)^2 + 5^2} + 3000\times \sqrt{x^2 + 3^2}[/tex]
We differentiate the above equation and equate it to zero to get the minimum cost as
[tex]\frac{\mathrm{d} (7000\times \sqrt{(10-x)^2 + 5^2} + 3000\times \sqrt{x^2 + 3^2})}{\mathrm{d} x}[/tex] = 0
[tex]7000\frac{2x-20}{2\sqrt{x^2-20x+125} } +3000\frac{2x}{2\sqrt{x^2+9} } = 0[/tex]
[tex]3500\frac{2x-20}{\sqrt{x^2-20x+125} } = -1500\frac{2x}{\sqrt{x^2+9} }[/tex]
[tex]3500\frac{\sqrt{x^2+9}}{\sqrt{x^2-20x+125} } = -1500\frac{2x}{2x-20 }[/tex]
[tex]x^4-20x^3+10.54x^2-22.05x+110.25 =0[/tex]
Solving the quartic equation we get
x = 7.82 miles
Therefore the length of is given as
Length of pipe to Chesterville [tex]\sqrt{7.82^{2} +3^2 } = 8.376 \, miles[/tex]
Length of pipe to Denton = [tex]\sqrt{(10-7.82)^2 + 5^2} = 5.46 \, miles[/tex].
