Question

for a given week, rachels coffee house has available 600 ounces of a grade coffee and 875 ounces of b grade coffee. these are blended into packages as follows: a deluxe blend that contains 3 ounces of a grade coffee and 7 ounces of b grade coffee, and a special blend that contains 4 ounces of a grade coffee and 5 ounces of b grade coffee. let x be the number of deluxe blend packages sold. let y be the number of special blend packages sold. shade the region corresponding to all values of x and y that satisfy these requirements.

Explanation:

Step1: Define constraints for A grade coffee

The total A grade coffee used in deluxe and special blends must be ≤ 600 ounces. So, \( 3x + 4y \leq 600 \).

Step2: Define constraints for B grade coffee

The total B grade coffee used in deluxe and special blends must be ≤ 875 ounces. So, \( 7x + 5y \leq 875 \).

Step3: Non - negative constraints

Since the number of packages can't be negative, \( x\geq0 \) and \( y\geq0 \).

To graph these inequalities:

1. For \( 3x + 4y=600 \), when \( x = 0 \), \( y=\frac{600}{4} = 150 \); when \( y = 0 \), \( x=\frac{600}{3}=200 \). Draw the line and shade the region below (since \( 3x + 4y\leq600 \)) and \( x\geq0,y\geq0 \).

1. For \( 7x + 5y = 875 \), when \( x=0 \), \( y=\frac{875}{5}=175 \); when \( y = 0 \), \( x=\frac{875}{7} = 125 \). Draw the line and shade the region below (since \( 7x + 5y\leq875 \)) and within the region defined by \( x\geq0,y\geq0 \).

The feasible region is the intersection of the regions defined by \( 3x + 4y\leq600 \), \( 7x + 5y\leq875 \), \( x\geq0 \), and \( y\geq0 \).

Answer:

The feasible region is bounded by the lines \( 3x + 4y = 600 \), \( 7x + 5y = 875 \), \( x = 0 \), and \( y = 0 \). To shade the region, first graph \( 3x+4y\leq600 \) (shade below the line \( 3x + 4y=600 \) in the first quadrant), then graph \( 7x + 5y\leq875 \) (shade below the line \( 7x + 5y = 875 \) in the first quadrant), and the overlapping region (including the boundaries) is the region of all \((x,y)\) that satisfy the requirements.