Concrete mix, which is used in jobs as varied as making sidewalks and building bridges, is composed of five main materials: cement, water, sand, gravel, and fly ash. By varying the percentages of these materials, mixes of concrete can be produced with differing characteristics. For example, the water-to-cement ratio affects the strength of the final mix, the sand-to-gravel ratio affects the "workability" of the mix, and the fly-ash-to-cement ratio affects the durability. Since different jobs require concrete with different characteristics, it is important to be able to produce custom mixes.

Assume you are the manager of a building supply company and plan to keep on hand three basic mixes of concrete from which you will formulate custom mixes for your customers. The basic mixes have the following characteristics:

	Super-Strong Type S	All-Purpose Type A	Long-Life Type L
Cement	20	18	12
Water	10	10	10
Sand	20	25	15
Gravel	10	5	15
Fly ash	0	2	8

Each measuring scoop of any mix weighs 60g, and the numbers in the table above give the breakdown by grams of the components of the mix. Custom mixes are made by combining the three basic mixes. For example, a custom mix might have 10 scoops of Type S, 14 of Type A, and 7.5 of Type L. We can represent any mixture by a vector [c,w,s,g,f] in R⁵ representing the amounts of cement, water, sand, gravel, and fly ash in the final mix. The basic mixes can therefore be represented by the following column vectors:

S=[20,10,20,10,0].
A=[18,10,25,5,2]
L=[12,10,15,15,8]

Part A: Give a practical interpretation to the linear combination 3S+5A+2L by discussing the resulting strength (low water to cement ratio), workability (high sand to gravel ratio), and durability (high fly ash to cement ratio) of the mix.

Part B: What does Span{S,A,L} = {a S + b A + c L where a, b, and c are real numbers} represent?

Part C: A customer requests 6 kg of (6000 g) of a custom mix with the following proportions of cement, water, sand, gravel, and fly ash: 16:10:21:9:4. Find the amounts of each of the basic mixes (S, A, and L) needed to create this mix.

Part D: Is the solution unique? Explain.