Linear programming
A candy merchant sells two variety bags of cookies. Each pound of variety X contains 60 % chocolate chips and 40% raisin bran and sells for $ 8.00 a pound. Each pound of variety Y contains 45% of chocolate chips and 55% raisin bran and sells for $ 10 .00 a pound. The merchant has available 400 pounds of chocolate chips and 300 pounds of raisin bran. The merchant will try to sell the amount of each blend that maximizes her income. Let x be the number of pounds of variety bag X and y be the number of pounds of variety bag Y.
(a). Since the merchant above has available 300 pounds of raisin bran , what inequality must be satisfied in the situation above
(b) What is the objective function?
cc = .6 x + .45 y </= 400
rb = .4 x + .55 y </= 300
income = i = 8 x + 10 y objective, maximize
graph those functions, check the corners
http://www.zweigmedia.com/RealWorld/LPGrapher/lpg.html
maximize p = 8x + 10y subject to
.6x + .45y <= 400
.4x + .55y <= 300
===============================
Vertex Lines Through Vertex Value of Objective
(566.666667,133.333333) .6x+.45y = 400; .4x+.55y = 300 5866.666667 Maximum
(666.666667,0) .6x+.45y = 400; y = 0 5333.333333
(0,545.454545) .4x+.55y = 300; x = 0 5454.545455
(0,0) x = 0; y = 0 0
Damon,
Thank you! Thank you so much! You are such a blessing to Math. :)
Abby
(a) In order to determine the inequality that must be satisfied for the situation above, we need to consider the available resources. The merchant has 300 pounds of raisin bran available.
Each pound of variety X contains 40% raisin bran, so the total pounds of raisin bran used for variety X would be 0.4x.
Each pound of variety Y contains 55% raisin bran, so the total pounds of raisin bran used for variety Y would be 0.55y.
Therefore, the total pounds of raisin bran used for both varieties would be 0.4x + 0.55y.
Since the merchant has available 300 pounds of raisin bran, the inequality that must be satisfied is:
0.4x + 0.55y ≤ 300
(b) The objective function represents the goal that the merchant is trying to maximize, which in this case is her income.
Let's calculate the income for each variety:
Income from variety X = $8.00 x (number of pounds of variety X) = $8.00x
Income from variety Y = $10.00 x (number of pounds of variety Y) = $10.00y
To maximize her income, the objective function is:
Maximize:
Income = $8.00x + $10.00y