2.2.20    Given the following information:

·        Car requires 1 ton of steel and 0.5 tons of plastic

·        Airplane requires 5 tons of steel and 2 tons of plastic

·        Steel consumes 3 tons of bituminous coal and 20 barrels of water

·        Plastic consumes 2 tons of coal and 50 barrels of water

Organize these facts into matrix form and find the matrix that shows how much coal (b) and water (w) is needed to make “c” cars and “a” airplanes

Define Variables:

Define Given Information into Matrices:

The quantities of steel and plastic required to construct (1) car and (1) airplane can be represented by the following matrix:

A car requires 1 ton of steel and 0.5 tons of plastic to build and a airplane requires 5 tons of steel and 2 tons of Plastic.

The quantities of bituminous coal and water consumed to make 1 ton of steel and 1 ton of plastic can be represented by the following matrix:

1 ton of steel requires 3 tons of coal and 20 barrels of water and 1 ton of plastic requires 2 tons of coal and 50 barrels of water.

Derive equation for amount of coal and water required to construct “c” cars and “a” airplanes:

Using the above definitions of variables and matrices we can express the amount of steel and plastic required in the following equation:

(1)

Similarly, we can represent the amount of coal and water in terms of steel and plastic with the following equation:

(2)

Substituting equation (1) into equation (2):

(3)

Using matrix multiplication:

(4)

Substituting equation (4) back into equation (3) gives the equation for finding the amount of coal and barrels of water required for “c” cars and “a” airplanes:

Similarly we can rewrite this matrix into algebraic form: