Question

A grocer buys cases of almonds and walnuts. Almonds are packaged 20 bags per case. The grocer pays $30 per case of almonds and makes a profit of $17 per case. Walnuts are packaged 24 bags per case.The grocer pays $26 per case of walnuts and makes a profit of $15 per case. He orders no more than 300 bags of almonds and walnuts together at a maximum "cost" of $400.

Answer 1

a. Write the constraints. Use X for the # of cases of almonds ordered and Y for the number of cases of walnut
b. Graph the constraints
c. Write the objective function for the profit
d. How many cases of almonds and walnuts maximize the grocer's profit?

Answer 2

I'm very confused on how to write my constraints.

Answer 3

you start by using
Use X for the # of cases of almonds ordered and Y for the number of cases of walnut

x is the number of cases of almonds
each case of almonds has 20 bags.

Now consider: if you had 1 case of almonds, you would have 20 bags
if you had 2 cases, then 40 bags
or 3 cases, then 60 bags
notice we do: number of cases times 20
use that rule, but use x as the "number" of cases

can you write how many bags are in x cases of almonds?

Answer 4

Is it 20x?

Answer 5

yes. and there are 24y bags in y cases of almonds

Answer 6

if we add up all the bags, we need
no more than 300 bags

any idea how to write that ?

Answer 7

x+y≤300?

Answer 8

almost. but x is the number of *cases* of almonds, and y is cases of walnuts

we want *bags* (which we figured out up above)

Answer 9

Oh! So it's 20x+24y≤300?

Answer 10

yes
and if we write it in "slope-intercept form"
y <= (-20/24)y +300/24
y <= (-5/6)y + 12.5
which we can plot

Answer 11

I'm a bit confused... Why is there 2 y's for each equation?

Answer 12

oh, bad typing. it should be
y <= (-5/6)x + 12.5

Answer 13

Do I plug in my answers after that? Or is there another equation?

Answer 14

this is a complicated question (lots of steps).

let's do the cost.

pays $30 per case of almonds (x)
pays $26 per case of walnuts (y)
and idea how to write the total cost for x cases of almonds and y cases of walnuts ?

Answer 15

it is almost the same idea as for the number of bags

Answer 16

30x+26y≤400?

Answer 17

yes, do you have a way to plot these ?

Answer 18

By using a graph?

Answer 19

we did part a
20x+24y≤300
30x+26y≤400

now part b says plot them. So you have to use either
graph paper or a computer

Answer 20

There is a piece of graph paper on the worksheet. It's numbered up until 20 on both the x and y axis, but I don't have a picture of it.

Answer 21

ok, it should look like this

Answer 22

Oh! Okay, thank you! So what do I do after this?

Answer 23

now write down the profit
same idea as the constraints

Answer 24

Does that mean the points on the graph?

Answer 25

makes a profit of $17 per case of almonds (x)
$15 per case of walnuts (y)

Answer 26

17x+15y=P?

Answer 27

yes
now to find the best profit, you test the points on the graph
(the "corners")
for example, test (0,12.5). The profit is 17*0 +15*12.5= 187.50
test the other points, and pick the one with the best profit.

Answer 28

(9, 5)
17*9+15*5=228

(13.33, 0)
17*13.33+ 15*0=226.61

Is (9, 5) the answer?

Answer 29

looks like it
so 9 cases of x (almonds) and 5 cases of y (walnuts)

Answer 30

So the maximum profit is 9 cases of almonds and 5 cases of walnuts? And that's the end?

Answer 31

Thank you so much!! I wish I could give you 100,000,000 medals for helping me for so long!