Find the optimal solution for the following problem. (round your answers to 3 decimal places.)
maximize c = 9x + 7y
subject to 8x + 10y ≤ 17
11x + 12y ≤ 25
and x ≥ 0, y ≥ 0.
1. what is the optimal value of x?
2. what is the optimal value of y?
3. what is the maximum value of the objective function?

There are several ways to calculate optimal solution; one of them, is by using graphs.

The given parameters are:

See attachment for the graphs of and

From the graph, the feasible regions are:

Test these values in the objective function

So, the value of C is maximum at:

So, we have:

Read more about maximizing functions at:

link

Maximize C =

and x ≥ 0, y ≥ 0

Plot the lines on graph

So, boundary points of feasible region are (0,1.7) , (2.125,0) and (0,0)

Substitute the points in  Maximize C

At  (0,1.7)

Maximize C =

Maximize C =

At  (2.125,0)

Maximize C =

Maximize C =

At   (0,0)

Maximize C =

Maximize C =

So, Maximum value is attained at   (2.125,0)

So, the optimal value of x is 2.125

The optimal value of y is 0

The maximum value of the objective function is 19.125

step by step resolutions

1) n+n+1+n+2=51

3n+3=51

3n=48

n=16

the three numbers are 16, 17 and 18.

2) x+15=4x

3x=15

x=5

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3) the two digits are x and y.

y=3x

(10x+y)+(10y+x)=88

y=3x

11x+11y=88

y=3x

x+y=8

x+3x=8

x=2

the original number can be either 28 or 82.

4) n+(n+2)+(n+4)=48

3n+6=48

3n=42

n=14

the numbers are 14, 16 and 18.

5) y=x+3

10y+x+10x+y=143

y=x+3

11x+11y=143

y=x+3

x+y=13

x+x+3=13

2x=10

x=5

the original number can be 57 or 75.

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