Graph the feasibility region represented by the given inequalities and find the maximum profit if p = 4x + 5y. x ≥ 5 y ≤ 10 2x - y ≤ 8

First, graph the three inequalities:  x ≥ 5  ,  y ≤ 10  ,  2x - y  ≤ 8

Next, look for the 3 intersections (vertices) of the lines: (5, 2)  ,  (5, 10)   ,  (9, 10)

Finally, input the vertex coordinates into P = 4x + 5y and look for the biggest #.

(5, 2):  4(5) + 5(2)  = 20 + 10  = 30

(5, 10): 4(5) + 5(10)  = 20 + 50  = 70

(9, 10): 4(9) + 5(10)  = 36 + 50  = 86

maximum profit = 86

step-by-step explanation:

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!