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Chapter 4: Problem 3

Suppose the objective function is \(Q=x^{2} y\) and you know that \(x+y=10 .\) Write the objective function first in terms of \(x\) and then in terms of \(y\).

Short Answer

Expert verified
Question: Express the objective function \(Q = x^2y\) in terms of \(x\) only and then in terms of \(y\) only, given the constraint \(x + y = 10\). Answer: The objective function \(Q\) is expressed in terms of \(x\) as \(Q = 10x^2 - x^3\) and in terms of \(y\) as \(Q = 100y - 20y^2 + y^3\).

Step by step solution

01

Solve for \(y\) in terms of \(x\) from the constraint

Using the constraint \(x + y = 10\), we solve for \(y\). \[y = 10 - x\] 2. Substitute \(y\) into the objective function
02

Substitute the expression for \(y\) into the objective function

Now that we have \(y\) in terms of \(x\), we can substitute it into the objective function: \[Q = x^2(10-x)\] 3. Simplify the objective function
03

Simplify the expression for the objective function in terms of \(x\)

Expanding the expression, we get: \[Q = 10x^2 - x^3\] 4. Express the constraint in terms of \(y\) only
04

Solve for \(x\) in terms of \(y\) from the constraint

Using the constraint \(x + y = 10\), we solve for \(x\). \[x = 10 - y\] 5. Substitute \(x\) into the objective function
05

Substitute the expression for \(x\) into the objective function

Now that we have \(x\) in terms of \(y\), we can substitute it into the objective function: \[Q = (10-y)^2y\] 6. Simplify the objective function
06

Simplify the expression for the objective function in terms of \(y\)

Expanding and simplifying the expression, we get: \[Q = 100y - 20y^2 + y^3\] Now we have expressed the objective function \(Q\) first in terms of \(x\) as \[Q = 10x^2 - x^3\] and then in terms of \(y\) as \[Q = 100y - 20y^2 + y^3\]

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