Answer:
The minimum cost of installing the cable is $156.
Step-by-step explanation:
We have an optimization problem.
We have to minimize the cost of the cable.
We will use the variable x to express the the length of cable CP and PM, accordingly to the attache picture.
The length of the cable that goes from C to P (let's call it CP) is x.
[tex]\bar{CP}=x[/tex]
Then, the length of the cable that goes from P to M (PM) can be calcualted usign the Pithagorean theorem:
[tex]\bar{PM}=\sqrt{(33-x)^2+9^2}[/tex]
The cost function Y is:
[tex]Y=4*\bar{CP}+5*\bar{PM}=4x+5\sqrt{(33-x)^2+9^2}[/tex]
To optimize this cost funtion we have to derive and equal to 0:
[tex]\dfrac{dY}{dx}=0\\\\\\\dfrac{dY}{dx}=4+5(\dfrac{1}{2})((33-x)^2+9^2)^{-1/2} *(-2)(33-x)\\\\\\\dfrac{dY}{dx}=4+5\dfrac{x-33}{\sqrt{(33-x)^2+81}}=0\\\\\\\dfrac{x-33}{\sqrt{(33-x)^2+81}}=-\dfrac{4}{5}\\\\\\(x-33)=-\dfrac{4}{5}\sqrt{(33-x)^2+81}\\\\\\(x-33)^2=(-\dfrac{4}{5})^2[(x-33)^2+81]\\\\\\(x-33)^2=\dfrac{16}{25}(x-33)^2+\dfrac{1296}{25}\\\\\\\dfrac{25-16}{25} (x-33)^2=\dfrac{1296}{25}\\\\\\9(x-33)^2=1296\\\\\\x-33=\sqrt{\dfrac{1296}{9}}=\sqrt{144}=\pm12\\\\\\x=33\pm12\\\\\\x_1=33-12=21\\\\x_2=33+12=45[/tex]
The valid solution is x=21, as x can not phisically larger than 33.
The cost then becomes:
[tex]Y=4*\bar{CP}+5*\bar{PM}=4x+5\sqrt{(33-x)^2+9^2}\\\\\\Y=4*21+5\sqrt{(33-21)^2+81}\\\\Y=81+5\sqrt{144+81}\\\\Y=81+5\sqrt{225}\\\\Y=81+5*15\\\\Y=81+75\\\\Y=156[/tex]
