Question: Find the minimum value of the function \( z = 2x + 2y \) subject to the following constraints…

Find the minimum value of the function \( z = 2x + 2y \) subject to the following constraints.

\[ \begin{cases} x \leq 17 \\ y \leq 16 \\ 5x + 2y \geq 42 \\ 3x + 11y \geq 84 \end{cases} \]

Note that the ALEKS graphing calculator can be used to make computations easier.

\( z = \boxed{\phantom{0}} \)

Solution

To find the minimum value of the function \( z = 2x + 2y \) subject to the given constraints, we first identify the feasible region defined by the constraints: \[ \begin{cases} x \leq 17 & \\ y \leq 16 & \\ 5x + 2y \geq 42 & \\ 3x + 11y \geq 84 & \end{cases} \] Next, we find the vertices of the feasible region by solving the system of inequalities. The intersection points of the lines will be considered since these points could potentially give the minimum value for \( z \). 1. Intersection of \( 5x + 2y = 42 \) and \( x = 17 \): Substitute \( x = 17 \) into \( 5x + 2y = 42 \): \[ 5(17) + 2y = 42 \] \[ 85 + 2y = 42 \] \[ 2y = 42 - 85 \] \[ 2y = -43 \Rightarrow y = -\frac{43}{2} \] This point is outside the feasible region since \( y \) must be less than or equal to 16. 2. Intersection of \( 5x + 2y = 42 \) and \( y = 16 \): Substitute \( y = 16 \) into \( 5x + 2y = 42 \): \[ 5x + 2(16) = 42 \] \[ 5x + 32 = 42 \] \[ 5x = 10 \] \[ x = 2 \] The point is \( (2, 16) \). 3. Intersection of \( 3x + 11y = 84 \) and \( y = 16 \): Substitute \( y = 16 \) into \( 3x + 11y = 84 \): \[ 3x + 11(16) = 84 \] \[ 3x + 176 = 84 \] \[ 3x = 84 - 176 \] \[ 3x = -92 \Rightarrow x = -\frac{92}{3} \] This point is outside the feasible region since \( x \) must be greater than or equal to 0. 4. Intersection of \( 5x + 2y = 42 \) and \( 3x + 11y = 84 \): Solve the two equations: Multiply the first by 3: \[ 15x + 6y = 126 \] Multiply the second by 5: \[ 15x + 55y = 420 \] Subtract the first from the second: \[ 49y = 294 \Rightarrow y = 6 \] Substitute \( y = 6 \) into \( 5x + 2y = 42 \): \[ 5x + 2(6) = 42 \] \[ 5x + 12 = 42 \] \[ 5x = 30 \Rightarrow x = 6 \] The point is \( (6, 6) \). Now, evaluate \( z = 2x + 2y \) at the feasible points: \( (2, 16) \) and \( (6, 6) \). For \( (2, 16) \): \[ z = 2(2) + 2(16) = 4 + 32 = 36 \] For \( (6, 6) \): \[ z = 2(6) + 2(6) = 12 + 12 = 24 \] The minimum value is \( 24 \) at the point \( (6, 6) \).