A plane delivers two types of cargo between two destinations. Each crate of cargo I is 2 cubic feet in volume and 73 pounds in weight, and earns $15 in revenue. Each crate of cargo II is 2 cubic feet in volume and 146 pounds in weight, and earns $45 in revenue. The plane has available at most 130 cubic feet and 5,548 pounds for the crates. Finally, at least twice the number of crates of I as II must be shipped. Find the number of crates of each cargo to ship in order to maximize revenue. Find the maximum revenue.

First of all, we note that Cargo II crates earn more revenue per cubit foot and more revenue per pound than Cargo I crates, so we want to maximize the shipment of Cargo II.

Since 2 crates of Cargo I must be delivered for each crate of Cargo II, it is convenient to bundle them together. Such a bundle will weigh ...

2×73 + 146 = 292 . . . . pounds

and will have a volume of ...

2×2 + 1×2 = 6 . . . . ft³

Then the number of bundles allowed by the volume limit is ...

(130 ft³)/(6 ft³) = 21 2/3

and the number of bundles allowed by the weight limit is ...

(5548 pounds)/(292 pounds) = 19

The weight limit is more restrictive, and the number of bundles it allows is an integer, so the load limit will be filled with 19 Cargo II and 38 Cargo I crates. Even though there is extra volume available, the weight limit will not allow any more cargo.

The revenue for each bundle is ...

2×$15 +1×$45 = $75

so the maximum revenue is 19×$75 = $1425.