Question

What is the maximum value of the objective function, P, with the given constraints?


P=3.75x+4.25y


2x+3y≤60

2x+y≤28

4x+y≤48

x≥0

y≥0




Enter your answer in the box.


P =

Answer 1

Answer:

The maximum value of the objective function is $P=90.5$

Step-by-step explanation:

$2x+3y\leq 60$ -----> constraint A

$2x+y\leq 28$ -----> constraint B

$4x+y\leq 48$ -----> constraint C

$x\geq 0$ ------> constraint D

$y\geq 0$ ------> constraint E

using a graphing tool

The solution of the constraints is the shaded area

see the attached figure

The vertices of the shaded area are

$A(0,0),B(0,20),C(6,16),D(12,0)$

Substitute the value of x and the value of y of each vertices in the objective function to determine the maximum value

we have

$P=3.75x+4.25y$

so

1) For point $A(0,0)$

$P=3.75(0)+4.25(0)=0$

2) For point $B(0,20)$

$P=3.75(0)+4.25(20)=85$

3) For point $C(6,16)$

$P=3.75(6)+4.25(16)=90.5$

4) For point $D(12,0)$

$P=3.75(12)+4.25(0)=45$

therefore

The maximum value of the objective function is $P=90.5$