A home-based company produces both hand-knitted scarves and sweaters.
please help please
The scarves take 2 hours of labor to produce, and the sweaters take 14 hours. The labor available is limited to 40 hours per week, and the total production capacity is 5 items per week. Write a system of inequalities representing this situation, where x is the number of scarves and y is the number of sweaters. Then graph the system of inequalities
2x + 14y </= 40
You can also write that as
x + 7y </= 20
x + y <= 5
Draw both of these straight lines:
y = 20/7 - x/7
y = 5-x
Then find the region of the graph that lies below BOTH lines, AND for which both x and y are zero or positive. It is a quadrilateral with a right-angle corner at the origin. We cannot draw graphs for you here.
Make scarves =x and sweaters = y.
2x + 14y <(or equal to) 40.
I think that's one of the constraints but I'm not sure how to get the other one.
I don't think my answer is right though!
I'm stuck on this 1 math problem. I've tried many ways to figure it out, but I can't. The problem is; 2(3x-5)>2x+6.
To represent the given situation in the form of a system of inequalities, we need to consider the constraints for labor and production capacity.
1. Labor Constraint:
The labor available is limited to 40 hours per week. As given, scarves take 2 hours of labor each, and sweaters take 14 hours. Thus, the labor constraint can be stated as:
2x + 14y ≤ 40
2. Production Capacity Constraint:
The total production capacity is 5 items per week. So, the production capacity constraint is:
x + y ≤ 5
Therefore, the system of inequalities representing the situation is:
2x + 14y ≤ 40
x + y ≤ 5
Now, let's graph these inequalities:
First, we graph the boundary lines for each inequality:
1. Graphing 2x + 14y ≤ 40:
a. Rewrite the inequality in slope-intercept form (y = mx + b):
14y ≤ -2x + 40
y ≤ (-2/14)x + (40/14)
y ≤ (-1/7)x + (20/7)
Plot the boundary line with equation y = (-1/7)x + (20/7), using the slope (-1/7) and y-intercept (20/7).
2. Graphing x + y ≤ 5:
a. Rewrite the inequality in slope-intercept form (y = mx + b):
y ≤ -x + 5
Plot the boundary line with equation y = -x + 5. This line has a slope of -1 and y-intercept at (0, 5).
Once you have plotted both the boundary lines, shade the region below the line y = (-1/7)x + (20/7) and below the line y = -x + 5 to represent both inequalities. The shaded region will represent the feasible region, satisfying both the labor and production capacity constraints.
Additionally, the region of interest will lie within the first quadrant since the number of scarves and sweaters cannot be negative.
I hope this helps!