A manufacturer produces a 4-cup and 8-cup coffee maker. The 4-cup maker takes 6 hours to produce and the 8-cup takes 9 hours. The manufacturer has at most 500 hours of labor per week.
a. Write an inequality to represent the number of each type of coffee makers they can produce in a week.
b. Is it possible to produce 20 4-cup and 30 8-cup coffee makers in a given week? Explain why or why not showing all of your calculations.
This is only the first step, linear programming is on your horizon :)
f for four cup
e for eight cup
6 f + 9 e </= 500
6 (20) + 9 (30) = 390 so sure we can do it
To answer part (a), we need to find the maximum number of each type of coffee maker the manufacturer can produce within the given labor constraint.
Let's assume the number of 4-cup coffee makers produced in a week is represented by 'x' and the number of 8-cup coffee makers produced in a week is represented by 'y'.
The time it takes to produce one 4-cup coffee maker is 6 hours, so 'x' 4-cup coffee makers will take 6x hours.
Similarly, the time it takes to produce one 8-cup coffee maker is 9 hours, so 'y' 8-cup coffee makers will take 9y hours.
The manufacturer has at most 500 hours of labor per week, so the inequality representing this constraint is:
6x + 9y ≤ 500
Now, let's move on to part (b) to see if it is possible to produce 20 4-cup and 30 8-cup coffee makers in a given week.
We substitute the values into our inequality:
6x + 9y ≤ 500
6(20) + 9(30) ≤ 500
120 + 270 ≤ 500
390 ≤ 500
The inequality is true, which means it is possible to produce 20 4-cup and 30 8-cup coffee makers in a given week within the labor constraint.
The calculations show that the total labor required for 20 4-cup coffee makers is 120 hours (6 hours per coffee maker), and the total labor required for 30 8-cup coffee makers is 270 hours (9 hours per coffee maker). The sum of these labor hours is 390, which is less than or equal to the maximum labor of 500 hours. Therefore, it is possible to produce the desired number of coffee makers within the given labor constraint.