A manufacturer has 150 kg and 90 kg of wood and plastic respectively. Product A require 1kg of wood and 2kg of plastic. Product B requires 3kg of wood and 1kg of plastic. If A sells for #4000 and B for #6000. how many of each should be made to obtain the maximum gross Income?
Let's assume the manufacturer makes x units of Product A and y units of Product B.
For each unit of Product A, the manufacturer needs 1 kg of wood and 2 kg of plastic.
Therefore, the total amount of wood required for x units of Product A is 1 kg × x units = x kg.
For each unit of Product B, the manufacturer needs 3 kg of wood and 1 kg of plastic.
Therefore, the total amount of wood required for y units of Product B is 3 kg × y units = 3y kg.
The total amount of wood used cannot exceed the available wood, which is 150 kg.
Therefore, x ≤ 150 kg and 3y ≤ 150. We can simplify this to y ≤ 50 kg.
For each unit of Product A, the manufacturer needs to use 2 kg of plastic.
Therefore, the total amount of plastic required for x units of Product A is 2 kg × x units = 2x kg.
For each unit of Product B, the manufacturer needs to use 1 kg of plastic.
Therefore, the total amount of plastic required for y units of Product B is 1 kg × y units = y kg.
The total amount of plastic used cannot exceed the available plastic, which is 90 kg.
Therefore, 2x + y ≤ 90 kg.
The gross income from selling x units of Product A is 4000 * x = 4000x.
The gross income from selling y units of Product B is 6000 * y = 6000y.
To obtain the maximum gross income, we need to maximize the expression 4000x + 6000y.
To solve this problem, we can use linear programming or graphing techniques. However, given the small number of units and the simplicity of the problem, we can simply use trial and error.
Let's start with x = 0 and increase x while decreasing y by 1 until we reach the constraint of 2x + y ≤ 90 kg.
For x = 0, y = 90, the gross income is 4000 * 0 + 6000 * 90 = 540000.
For x = 1, y = 88, the gross income is 4000 * 1 + 6000 * 88 = 528000.
Continuing this process, we find that for x = 20, y = 50, the gross income is 4000 * 20 + 6000 * 50 = 640000.
We can't increase x any further because it will exceed the wood constraint of x kg ≤ 150 kg.
Therefore, to obtain the maximum gross income, the manufacturer should make 20 units of Product A and 50 units of Product B.