Sure, let's solve the problem step-by-step.
You want to determine the minimum cost to run machine [tex]\( X \)[/tex] and machine [tex]\( Y \)[/tex] to produce the required number of units of products [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
Given:
- Machine [tex]\( X \)[/tex] produces 4 units of [tex]\( A \)[/tex] and 5 units of [tex]\( B \)[/tex] per hour.
- Machine [tex]\( Y \)[/tex] produces 3 units of [tex]\( A \)[/tex] and 10 units of [tex]\( B \)[/tex] per hour.
- Cost to run machine [tex]\( X \)[/tex] is [tex]$22 per hour.
- Cost to run machine \( Y \) is $[/tex]25 per hour.
Let:
- [tex]\( x \)[/tex] be the number of hours machine [tex]\( X \)[/tex] runs.
- [tex]\( y \)[/tex] be the number of hours machine [tex]\( Y \)[/tex] runs.
The objective function [tex]\( C \)[/tex] representing the total cost is:
[tex]\[ C = 22x + 25y \][/tex]
Units produced per hour:
1. For [tex]\( A \)[/tex]:
[tex]\[ 4x + 3y \][/tex]
2. For [tex]\( B \)[/tex]:
[tex]\[ 5x + 10y \][/tex]
To determine the exact solution, we need the constraints related to required units of [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Assuming [tex]\( a \)[/tex] units of [tex]\( A \)[/tex] and [tex]\( b \)[/tex] units of [tex]\( B \)[/tex] are required, we have the following constraints:
1. [tex]\( 4x + 3y \geq a \)[/tex]
2. [tex]\( 5x + 10y \geq b \)[/tex]
3. [tex]\( x, y \geq 0 \)[/tex]
For simplicity, let's solve a hypothetical example where we need:
- [tex]\( a = 24 \)[/tex] units of [tex]\( A \)[/tex]
- [tex]\( b = 50 \)[/tex] units of [tex]\( B \)[/tex]
Then the constraints become:
1. [tex]\( 4x + 3y \geq 24 \)[/tex]
2. [tex]\( 5x + 10y \geq 50 \)[/tex]
Let's solve the system of inequalities:
### Step 1: Express [tex]\( x \)[/tex] in terms of [tex]\( y \)[/tex] from the second inequality:
[tex]\[ 5x + 10y \geq 50 \][/tex]
[tex]\[ 5x \geq 50 - 10y \][/tex]
[tex]\[ x \geq 10 - 2y \][/tex]
### Step 2: Substitute [tex]\( x \)[/tex] into the first inequality:
[tex]\[ 4(10 - 2y) + 3y \geq 24 \][/tex]
[tex]\[ 40 - 8y + 3y \geq 24 \][/tex]
[tex]\[ 40 - 5y \geq 24 \][/tex]
[tex]\[ -5y \geq -16 \][/tex]
[tex]\[ y \leq \frac{16}{5} \][/tex]
[tex]\[ y \leq 3.2 \][/tex]
So, our constraints are [tex]\( x \geq 10 - 2y \)[/tex] and [tex]\( y \leq 3.2 \)[/tex]. Let's substitute [tex]\( y = 3.2 \)[/tex]:
### Step 3: Find value of [tex]\( x \)[/tex]:
[tex]\[ x \geq 10 - 2(3.2) \][/tex]
[tex]\[ x \geq 10 - 6.4 \][/tex]
[tex]\[ x \geq 3.6 \][/tex]
### Step 4: To find minimum cost, let's find cost at [tex]\( y = 3.2, x = 3.6 \)[/tex]:
[tex]\[ C = 22x + 25y \][/tex]
[tex]\[ C = 22(3.6) + 25(3.2) \][/tex]
[tex]\[ C = 79.2 + 80 \][/tex]
[tex]\[ C = 159.2 \][/tex]
Therefore, the minimum cost to run the machines while meeting the production goals is:
[tex]\[ \$159.20 \][/tex]
This occurs when:
[tex]\[ x = 3.6 \text{ hours} \][/tex]
[tex]\[ y = 3.2 \text{ hours} \][/tex]
