127k views
0 votes
Find the maximum value of Z = 5x + 2y and subject to the constraints: 2x + 2y ≥ 6, x ≥ 0, y ≥ 0.

Find the maximum value of Z = 5x + 2y and subject to the constraints: 2x + 2y ≥ 6, x-example-1
User Neuman
by
8.1k points

1 Answer

6 votes

Looking at the constrains, we have that x and y must be positive values.

Also, if we divide the first inequality by two, we have that the sum of x and y must be greater than or equal 3.

Therefore, for the equation Z = 5x + 2y, the maximum value would be infinity, since the values of x and y can grow to any positive value and still satisfy the constrains.

If the question is asking "minimum value" instead of "maximum value", we can find it by using x = 0 and y = 3 (we choose the minimum possible value for x, because its coefficient in the equation Z = 5x + 3y is bigger, therefore it has the greater impact in the function value)

In this case, the minimum value of Z would be:


\begin{gathered} Z=5x+3y \\ x=0,y=3\colon \\ Z=5\cdot0+3\cdot3 \\ Z=9 \end{gathered}

There is no option for this value of Z.

So instead of calculating the minimum value, let's change the sign of the inequality 2x + 2y >= 6 for 2x + 2y <= 6.

In this case, we should maximize x, since it has the greater coefficient in the equation for Z.

So we should choose x = 3 and y = 0:


\begin{gathered} Z=5x+3y \\ x=3,y=0\colon \\ Z=5\cdot3+3\cdot0 \\ Z=15 \end{gathered}

Then, in this case, using the inequality 2x + 2y <= 6, the correct option would be the first one (15).

User Aeryaguzov
by
7.7k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.