Question

part b.
in one week, 380 units of product x are manufactured.
what is the greatest number of units of product y that can be manufactured that week?
options:
a) ( x + y geq 1000 )
( 3.5x + 2.5y leq 3000 )
b) ( x + y geq 1000 )
( x + y leq 3000 )
c) ( x + y leq 1000 )
( x + y geq 3000 )
d) ( x + y leq 1000 )
( 3.5x + 2.5y geq 3000 )

Explanation:

Step1: Identify the constraints

We know that \( x = 380 \) (units of Product X), and we need to use the relevant constraint from the options. From the problem, the cost - related constraint (since it has coefficients for \( x \) and \( y \)) is \( 3.5x + 2.5y\leq3000 \) (assuming this is the correct constraint for resource allocation or production limit).

Step2: Substitute \( x = 380 \) into the inequality

Substitute \( x = 380 \) into \( 3.5x+2.5y\leq3000 \). First, calculate \( 3.5\times380 \). \( 3.5\times380=(3 + 0.5)\times380=3\times380+0.5\times380 = 1140+190 = 1330 \).

Step3: Solve for \( y \)

The inequality becomes \( 1330+2.5y\leq3000 \). Subtract 1330 from both sides: \( 2.5y\leq3000 - 1330=1670 \). Then divide both sides by 2.5: \( y\leq\frac{1670}{2.5}=668 \). Also, we should check the other constraint \( x + y\geq1000 \). Substitute \( x = 380 \), we get \( 380 + y\geq1000\), so \( y\geq1000 - 380 = 620 \). But since we want the greatest number of units of \( y \), we focus on the upper - bound from the cost constraint.

Answer:

The greatest number of units of Product Y is 668.