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

The max value is z = 75

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

===========================================================

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 and ).

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

Lastly, we shade below the boundary 4x+3y = 60 because of the inequality . 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:

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

-----------------------

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).

take the logarithm of both sides (the base of the log doesn't matter):

apply the exponent property - this says that :

distribute the log terms on both sides:

group like terms together:

divide through both sides by the coefficient on :

you can do some additional rewriting here using the properties of the logarithm to simplify things if you like: