Optimization problem - minimize cost

In summary, the optimization problem is to minimize the cost of installing cable to control a motor floating on a buoy in a large pool. The cost includes both underwater and land cable costs, and the goal is to find the proportions of each type of cable that will minimize the cost. The solution involves setting up a function and using the chain rule to find the critical point.
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phantomcow2
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Optimization problem -- minimize cost

Homework Statement




A motor is floating on a buoy in a large pool with a perfectly flat shoreline. The motor is 3ft from the edge of the pool. A circuit to control operation of the motor is located 100ft from the closest point of the motor to the edge of the pool. Cable must be installed to control the motor. It costs 2.50/ft to run cable underwater, and 1.50/ft to run it on land. Find the proportions (amount of cable on land and amount on water) to minimize cost.


Homework Equations


Cost= (2.50w)+(1.50L) trying to minimize this function.
where w= underwater cable length
L= land cable length

w^2 = (100-L)^2 + 3^2 This is the "top" triangle that is formed. 3 is a fixed length, one side of the triangle. 10-L is the other side, and w is the hypotenuse.

The Attempt at a Solution



C=2.50[(100-L)^2+9]^.5 + 1.50L

Use chain rule to find C': (5/4)[(100-L)^2+9]^-.5(2L-200) + 3/2
Set equal to zero and solve for L...right
Rewriting this gives a rational function (because of the -1/2 power) plus a constant.

Is this correct thus far? I tried solving this by rewriting as a rational function. I know that I only need to concern myself with the numerator of rational functions to make the whole thing equal to zero, but I have that constant out there. I tried finding a common denominator, but then the only way I can make the function equal to zero is by setting length of land cable as zero.
 
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  • #2


phantomcow2 said:

The Attempt at a Solution



C=2.50[(100-L)^2+9]^.5 + 1.50L

Use chain rule to find C': (5/4)[(100-L)^2+9]^-.5(2L-200) + 3/2
Set equal to zero and solve for L...right
Rewriting this gives a rational function (because of the -1/2 power) plus a constant.

Is this correct thus far?

Yes, looks good so far.

Can you get the square-root term so it's on one side of the equation, all by itself? Then you'd just need to square everything to get rid of the radical.
 

What is an optimization problem?

An optimization problem is a mathematical problem that involves finding the best solution among a set of possible solutions. The goal is to minimize or maximize a certain objective function while satisfying a set of constraints.

What does it mean to minimize cost in an optimization problem?

Minimizing cost in an optimization problem means finding the solution that results in the lowest possible cost. This involves finding the optimal values for the variables in the problem that will result in the lowest overall cost.

How is an optimization problem with a cost objective function solved?

An optimization problem with a cost objective function is typically solved using mathematical techniques such as linear programming, dynamic programming, or calculus-based methods. These techniques involve finding the optimal values for the variables that will minimize the cost function while satisfying any constraints.

What are some common constraints in optimization problems?

Common constraints in optimization problems include resource constraints (such as limited budget or time), physical constraints (such as size or weight limitations), and technical constraints (such as production capacity or technical specifications).

What are some real-world applications of optimization problems with cost minimization?

Optimization problems with cost minimization have a wide range of real-world applications, including supply chain management, inventory control, production planning, transportation planning, and financial decision making. These problems can also be applied in various industries such as manufacturing, logistics, finance, and healthcare to improve efficiency and reduce costs.

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