What combination of oak and pine seedlings should Joey order to minimize shipping costs?

Question

Joey wants to order a combination of oak and pine seedlings. Joey wants at least 8 seedlings total and wants at least 5 to be pine seedlings. The box normally used for shipping can hold no more than 18 pounds. One oak seedling weighs 1 pound and costs $3 to ship. One pine seedling weighs 2 pounds but because of favorable shipping regulations only cost $2 to ship.

Answer 1

Let X represent Oak Seedlings and Y represent Pine Seedlings.

 

Cost of X is $3  and weighs 1 lb

Cost of Y is $2  and weighs 2 lb

Max shipping weight is 18 lb

 

Now, since we want  Y >= 5, then if Y= 5, the cost would be 5*$2 = $10 and weighs a total of 5*2= 10 lb

That means we have 8 lb left to use. Since X weighs 1 lb then we can buy 8X for a total of 8 lb.

The price for 8X is 8*$3 = $24.

This gives a total shipping cost of 10+24 = $34 where we purchase 5 Pine Seedlings and 8 Oak Seedlings.

 

Notice that since Y cost less even if it weighs more, then we can get away with purchasing more of Y while staying within the 18 lb limit.

 

We can repeat the above process and keep increasing Y (as long as we don't pass the 18lb limit) and we finally get:

 

8Y --> 8*2 = 16 lb at a price of  8*$2 = $16  and,

2X --> 2*1 = 2 lb at a price of 2*$3 = $6.

This way we can buy 8 Pine Seedlings for a Cost of $16 weighing 16 lb and 2 Oak Seedlings for a price of $6 weighing only 2 lb.

We get a total of 18lb, we purchase at least 5 Pine seedlings as needed ( in this case 8) and we purchase at least 8 seedlings total as requested ( in this case 10), all for a grand total of only $22 !