Answer:
question a and b are given in the explanation box.
(c). optimal solution z = 6900
Explanation:
(a). Given from the question that;
Resource Softball Baseball Available
Leather 5 oz 4 oz 6,000 oz
Nylon 6 yds 3 yds 5,400 yds
Core 4 oz 2 oz 4,000 oz
Labor 2.5 min 2 min 3,500 min
Stitching 1 min 1 min 1,500 min
let the Profit be
Softball : $17 - $11 = $6
Baseball : $15 - $10.4 = $4.5
also let x = number of softballs
and y = number of baseballs
The Linear Programming model for this becomes;
⇒ Maximize the Profit z = 6x + 4.5y
subject to the following constraints:
Leather : 5x + 4y ≤ 6000
Nylon : 6x + 3y ≤ 5400
Core : 4x + 2y ≤ 4000
Labor : 2.5x + 2y ≤ 3500
Sticthing : x + y ≤ 1500
(b). the next question tells us to sketch the feasible region, but first we will solve for the corresponding y and x values.
from the equation;
5x + 4y = 6000 .............(1)
and 6x + 3y = 5400 ...........(2)
equating (2) we have
x = 5400 - 3y / 6 ...........(3)
and from equation (1), we have
x = 6000 - 4y / 5 ........(5)
combining both value of x we have
5400 - 3y / 6 = 6000 - 4y / 5
where b7 elimination we have y = 1000
substituting value of y into equation (1), we have
5x + 4(1000) = 6000 = 400
x = 400
∴ y = 1000, x = 400
NB - the picture uploaded shows a diagrammatic sketch of the various regions with the shaded region seen
(c). given previously that z = 6x + 4.5y, we will compare each of the points (x, y)
thus,
at (0,1500), z = 6(0) + 4.5(1500) = 6750
at (900,0), z = 6(900) + 4.5(0) = 5400
at (400, 1000), z = 6(400) + 4.5(1000) = 6900
As seen in the answer above, Points (400,1000) gives the maximum
∴ The optimal solution is z = 6900.
cheers i hope this helps.
