Jimmy\'s Co ee and Things is the premiere place for co ee and things. They sell

Question

Jimmy's Co ee and Things is the premiere place for co ee and things. They sell two
                    di erent blends of co ee, Tropic Blend and Exotic Blend. Tropic Blend is one-half Costa Rican beans
                    and one half Ethiopian beans. Exotic blend is one-quarter Costa Rican beans and three-quarters
                    Ethiopian beans. Pro t from Tropic Blend is $4.00 per pound, while pro t from Exotic Blend is $3.50
                    per pound. Each day Jimmy's Co ee and Things receives 200 pounds of Costa Rican beans and 330
                    pounds of Ethiopian beans. How many pounds of each blend should be prepared to maximize pro t                 

Explanation / Answer

We like the $4 profit per pound, but we also want to use up as much of the beams as possible to sell more pounds of product. Let's look at the extreme values.

If we use all the CR beans to make Tropic Blend, we will have 400 pounds of tropic blend and $1,600 profit. We will also have 130 pounds of E beans left over. That is maximizing the waste too. So $1,600 is the number to beat. Now let's look at the other extreme; we will use up all of the beans.

After we make x pounds of Tropic blend we want three times more E beans left than CR beans (If there are three times more, it will be in a 1/4 : 3/4 ratio, since 3/4 is three times 1/4

If we use x pounds of CR beans we have (200 - x) left over (to use in the Exotic blend)

if we use x pounds of E beans we have (330 - x) left over

and we want the left over to be three times more, so to make them equal, we need to multiply the little one by three. This give us the following equation

3*(200 - x) = (330 - x)

600 - 3x    = 330 - x

270            = 2x

135            = x

This means we make 135 + 135 = 270 lbs of TBlend for a profit of $4*270 = $1,080 plus

65 + (330 - 135) = 260 lbs of Ex Blend for a profit of $3.5*260 = $910

for a total of $1,990 profits.

So now this is the number to beat.

Notice that 330 - 135 = 195, and that 65 is 1/3 of 195

Normally, we would now try to come up with an equation and take the calculus derivative to find the maximum, but, just for fun, I calculated the profits if we used 136 lb of each to make TBlend

136 + 136 = 272 lbs of TBlend for a profit of $4*272 = $1,088

but now we can't use all of our E beans

64 + 192 = 256 for a profit of $3.50*256 = $896

this gives a total of $1,984 already less than the $1,990 profits when we used up every single bean. Since leaving three pounds of Ethiopean bean unused already results in lower combined profits, there is no need to try to find the sweet spot, we already found it at the extremum value of 135 lb of each to make Tropical Blend and all of the rest used in making Exotic Blend.

Maximum profit

$1,990

135 lb CR beans for Tr Blend, 65 lb CR beans for Ex Blend

135 lb Ethio beans for Tr Blend, 195 lb Eth beans for Ex blend.

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