Question 1071178: At the end of the month, after filling orders for its regular customers, Ghirardelli Chocolate Company has some Twilight Delight Chocolate pieces of individually wrapped candy and some Dark Chocolate with sea salt individually wrapped pieces of candy left over. The practice of the company has been to package a mixture of the two chocolates into 1-pound packages as follows: a saltier blend of 4 ounces of Twilight Delight Chocolate and 12 ounces of Dark chocolate with sea salt and another blend containing 8 ounces of each type of candy. A profit of $.30 per package is made on the saltier blend and $.40 per package on the other blend. This month, 120 pounds of Dark Chocolate with sea salt and 100 pounds of Twilight Delight Chocolate are left over. How many packages of each mixture should be prepared to achieve a maximum profit? (Remember that there are 16 ounces in a pound)
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! first of all, you want to convert everything to pounds or ounces.
i chose to convert all to pounds.
1 pound is equal to 16 ounces, therefore, divide the number of ounces by 16 and you get the number of pounds.
you are given that the saltier blend contains 4 ounces of twilight delight chocolate and 12 ounces of dark chocolate with salt.
translate to pounds and you get that the saltier blend contains .25 pounds of twilight delight chocolate and .75 pounds of dark chocolate with salt.
you are given that the other blend contains 8 ounces of twilight delight chocolate and 8 ounces of dark chocolate with salt.
translate to pounds and you get that the other blend contains .5 pounds of twilight delight chocolate and .5 pounds of dark chocolate with salt.
you are also given that the number of available pounds of the dark chocolate with salt is equal to 120 and that the number of available pounds of twilight delight chocolate is equal to 100.
let x = the number of pounds of the saltier blend and let y = the number of pounds of the other blend.
your blends are sold in 1 pound packages.
your profit is .3 per package of the saltier blend and .4 per package of the other blend.
this translates to .3 per pound of the saltier blend and .4 per pound of the other blend.
your profit will therefore = .3 * number of pounds of the saltier blend + .4 * number of pounds of the other blend.
algebraically, this becomes profit = .3x + .4y.
that becomes your objective function because that's what you want to maximize.
your constraints will be:
x >= 0
y >= 0
this is because the number of pounds of either product can't be negative.
.25x + .5y <= 100
this is because the number of pounds of twilight delight chocolate must be less than or equal to 100 pounds and this is composed of .25 pounds of chocolate delight for each pound of the saltier blend and .5 pounds of chocolate delight for each pound of the other blend.
.75x + .5y <= 120
this is because the number of pounds of dark chocolate with salt must be less than or equal to 120 pounds and this is composed of .75 pounds of dark chocolate with salt for each pound of the saltier blend and .5 pounds of dark chocolate with salt for each pound of the other blend.
your objective function is:
.3x + .4y = profit.
your constraint functions are:
x >= 0
y >= 0
.25x + .5y <= 100
.75x + .5y <= 120
using the www.desmos.com/calculator, you would graph the opposite inequalities as shown below:
x <= 0
y <= 0
.25x + .5y >= 100
.75x + .5y >= 120
the area of the graph that is NOT shaded is your region of feasibility.
you would then evaluate the objective function at each of these corner points.
your maximum profit will be at one of those corner points.
your graph will look like this:
at the point (0,200), your profit is .4 * 200 = 80.
at the point (40,180), your profit is .3 *40 + .4 * 180 = 84.
at the point (160,0), your profits is .3 * 160 = 48.
your maximum profit is at the point (40,180).
this means you sell 40 pounds of the saltier blend and 180 pounds of the other blend.
your constraints need to be met at the point (40,180).
x and y >= 0 are met because x = 40 and y = 180.
.25x + .5y <= 100 is met because .25*40 + .5*180 = 100.
.75x + .5y <= 120 is met because .75*40 + .5*180 = 120.
if you were to do this manually, you would graph the equalities and then find the area on the graph that satisfied the inequalities.
you would graph:
x = 0
y = 0
.25x + .5y = 100
.75x + .5y = 120
you would then find the area on the graph that satisfies the inequality.
the inequalities are:
x >= 0
y >= 0
.25x + .5y <= 100
.75x + .5y <= 120
in that particular case, your graph would look like this:
in this case, the area that IS shaded is your region of feasibility.
note also that most graphing software require you to solve for y before graphing.
for example:
in the equation of .25x + .5y = 100, you would have to solve for y first.
your equation would become y = (100 - .25x) / .5.
that's the equation that you would graph.
it's the same equation so you'll get the same graph, but desmos.com allows you to graph is as .25x + .5y = 100 while other graphing software require you to graph it as y = (100 - .25x) / .5.
in fact, in most graphing software, the y is assumed, so you would graph (100 - .25x) / .5.
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