Question

Find the maximum value of the function z=5x+2y subject to following constraints

Answer 1

Answer: The max value is z = 75

It occurs when (x,y) = (15,0)

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Step-by-step explanation:

We'll need to graph this system of inequalities. See below.

The blue region represents the set of all (x,y) points that satisfy every inequality mentioned.

The horizontal lines represent the boundaries y = 0 and y = 8. We shade between those horizontal lines (since
`y \ge 0` and
`y \le 8`).

Moreover, we also shade to the right of x = 5 due to the inequality
`x \ge 5`

Lastly, we shade below the boundary 4x+3y = 60 because of the inequality
`4x+3y \le 60`. You can use a test point like (0,0) to check to see it's in the shaded region for that particular inequality.

With all those conditions in mind, you should get the graph shown below. The corner points are what we're after. Those corner points are:

• A = (5, 0)
• B = (5, 8)
• C = (9, 8)
• D = (15, 0)

Plug each set of x,y coordinates into the z = 5x+2y function

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Plugging in the coordinates for point A gets us

z = 5x+2y

z = 5(5)+2(0)

z = 25

We'll compare this value later on.

Repeat for the coordinates of point B

z = 5x+2y

z = 5(5)+2(8)

z = 41

Repeat for point C

z = 5x+2y

z = 5(9)+2(8)

z = 61

Then finally point D

z = 5x+2y

z = 5(15)+2(0)

z = 75

We found the largest z value and it's z = 75. It occurs when (x,y) = (15,0).