MHB Jesus' question at Yahoo Answers regarding finding total cost function

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The discussion revolves around formulating the total cost function for a pipeline from an oil refinery to storage tanks across a river. The problem involves calculating the lengths of the pipeline under the river and over land, with specific costs associated with each type of terrain. Using Pythagorean theorem, the lengths are expressed in terms of the distance from point P to the storage tanks. The total cost function is derived by substituting the lengths into the cost formula, resulting in a function that incorporates the costs per kilometer for both land and underwater sections. The final expression for the total cost function is presented as a simplified equation.
MarkFL
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Here is the question:

Function Math Question?

I am having a problem solving this question, any help is appreciated!

An oil refinery is located on the north bank of a straight river that is 2 km wide. A pipeline is to be constructed from the refinery to a point P on the south bank of the river, and then along the river to storage tanks on the south side of the river 6 km east of the refinery. The cost of laying pipe is $400,000 per km over land, and $800,000 per km under the river. Express the total cost of the pipeline as a function of the distance from P to the storage tanks.

I have posted a link there to this topic so the OP can see my work.
 
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Hello Jesus,

I would first label some variables:

$$L_R$$ = the length of the pipeline under the river.

$$L_L$$ = the length of the pipeline over land. This is the variable with which we are to express the total cost function.

$$D$$ = the distance downriver the tanks are from the refinery.

$$W$$ = the width of the river.

$$C_R$$ = cost in dollars per unit length to lay pipe under the river.

$$C_L$$ = cost in dollars per unit length to lay pipe over the land.

Next, let's draw a diagram:

View attachment 1022

$R$ is the location of the refinery, and $T$ is the location of the tanks.

We see that by Pythagoras, we have:

$$L_R=\sqrt{\left(D-L_L \right)^2+W^2}$$

Now, the total cost is given by:

$$C=C_RL_R+C_LL_L$$

Substituting for $L_R$, we have:

$$C\left(L_L \right)=C_R\sqrt{\left(D-L_L \right)^2+W^2}+C_LL_L$$

Using the given data for the parameters:

$$C_R=800000,\,D=6,\,W=2,\,C_L=400000$$

we have:

$$C\left(L_L \right)=800000\sqrt{\left(6-L_L \right)^2+2^2}+400000L_L$$

Factoring and simplifying, we have:

$$C\left(L_L \right)=400000\left(2\sqrt{L_L^2-12L_L+40}+L_L \right)$$
 

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