Aaron own a shipping company. He plans to move into his new office which is near to the city centre. He needs some filing cabinets to organize his files. Cabinet x which costs RM100 per unit, requires 0.6 square meters of the floor space and can hold 0.8 cubic meters of files. Cabinet y which costs RM200 per unit, requires 0.8 square meters of the floor space and can hold 1.2 cubic meters of files. The ratio of the number of cabinet x to the number of cabinet y is not less than 2:3. Aaron has an allocation of RM1400 for the cabinets and the office has room for no more than 7.2 square meters.

��i) Using the given information,�
a) write the inequalities which satisfy all the above constraints,�
b)construct and shade the region that satisfies all the above constraints.
�ii)Using two different methods, fin the maximum storage volume
�iii) Aaron plans to buy cabinet x in a range of 4 to 9 units. Tabulate all the possible combinations of the cabinets that he can purchase. Calculate the cost of each combination.�
iv) If you were Aaron which combination would you choose? Justify your answer and give your reasons.

It is an optimization problem.

Here are some steps to solve the problem:
1. define variables (values that are to be found).
2. Reread the question, and translate each constraint into an inequality.
3. Graph the constraints and select the feasible region, using the variables as the axes.
4. Determine the coordinates of the vertices of the feasible region.
5. tabulate the storage volume for each point, and select the vertex having the maximum storage volume. Remember that the variables (number of filing cabinets) must be integers.
6. In (iv), you have to add additional constraints and then repeat steps 4 and 5.

Show how far you have gone with these steps and post if you encounter difficulties.

i) To solve this problem, let's first define our variables.

Let x be the number of cabinet x Aaron buys, and y be the number of cabinet y.
We can use the following equations and inequalities to satisfy the given constraints:

a) Inequalities:
1) Cost Constraint:
100x + 200y ≤ 1400 (Aaron's allocation for cabinets)
2) Floor Space Constraint:
0.6x + 0.8y ≤ 7.2 (Office's maximum floor space)

b) Constructing the region:
To construct a region that satisfies all the above constraints, we can plot the graph of these inequalities on a coordinate plane and shade the region where all the inequalities are true:

1) Cost Constraint:
By rearranging the inequality, we get:
100x + 200y ≤ 1400
This equation represents a straight line on the coordinate plane. To graph it, we can find two points, plot them, and draw a straight line passing through those points.

When x = 0, y = 7 (1400/200)
When y = 0, x = 14 (1400/100)
Plotting these two points (0, 7) and (14, 0), we can draw a line passing through them.

2) Floor Space Constraint:
By rearranging the inequality, we get:
0.6x + 0.8y ≤ 7.2
This equation also represents a straight line on the coordinate plane. Following the same steps as above, we can find two points and plot them.

When x = 0, y = 9 (7.2/0.8)
When y = 0, x = 12 (7.2/0.6)
Plotting these two points (0, 9) and (12, 0), we can draw a line passing through them.

Shading the region where both lines overlap represents the region that satisfies all the given constraints.

ii) To find the maximum storage volume using two different methods:

Method 1: Using Geometry
We can use the graph from part b to find the vertex of the feasible region (the point where the maximum storage volume occurs).

Method 2: Using Linear Programming
We can set up a linear programming problem and solve it using optimization techniques, such as the simplex method or graphical method.

iii) To calculate all the possible combinations of cabinets X and Y that Aaron can purchase, we can use the range of 4 to 9 units for cabinet X and find the corresponding values for cabinet Y.

x | y | Cost
-------------
4 | ? | ?
5 | ? | ?
6 | ? | ?
7 | ? | ?
8 | ? | ?
9 | ? | ?

To calculate the cost for each combination, we can use the given cost per unit.

iv) To choose the combination, we need to consider the cost, storage volume, and feasibility. Taking into account the cost constraint, storage volume, and any other relevant factors, we can make the best decision.