A manufacturer of downhill and cross-country skis reports that manufacturing time is 2 hours and 3 hours, respectively, per ski and that finishing time is 6 hours for each downhill and 7 hours for each cross-country ski. There are only 24 hours per week available for the manufacturing process and 60 hours for the finishing process. The average profit is
$
71
$71 for downhill ski and
$
69
$69 for cross-country ski. The manufacturer wants to know how many of each type of ski should be made to maximize the weekly profit.
Corner points of the feasible region:
To find the corner points of the feasible region, we need to determine the maximum number of skis that can be manufactured within the given constraints.
Let's assume that x represents the number of downhill skis and y represents the number of cross-country skis.
Manufacturing Constraints:
2x + 3y ≤ 24 (manufacturing time constraint)
6x + 7y ≤ 60 (finishing time constraint)
Non-negativity Constraints:
x ≥ 0 (cannot manufacture a negative number of downhill skis)
y ≥ 0 (cannot manufacture a negative number of cross-country skis)
To find the corner points, we need to solve these constraints pair-wise. Let's solve the manufacturing time constraint and the non-negativity constraints first:
For x = 0, we get y = 8 (feasible point: (0, 8))
For y = 0, we get x = 12 (feasible point: (12, 0))
Now, let's solve the finishing time constraint and the non-negativity constraints:
For x = 0, we get y = 8.571 (feasible point: (0, 8.571))
For y = 0, we get x = 10 (feasible point: (10, 0))
The corner points of the feasible region are: (0, 8), (12, 0), (0, 8.571), and (10, 0).
To determine the corner points of the feasible region, we need to set up the constraints for the manufacturing and finishing processes.
Let's assume that the number of downhill skis produced is x and the number of cross-country skis produced is y.
Manufacturing Constraints:
2x + 3y ≤ 24 (Since there are 24 hours available for manufacturing)
x ≥ 0 (We can't produce a negative number of downhill skis)
y ≥ 0 (We can't produce a negative number of cross-country skis)
Finishing Constraints:
6x + 7y ≤ 60 (Since there are 60 hours available for finishing)
x ≥ 0 (We can't finish a negative number of downhill skis)
y ≥ 0 (We can't finish a negative number of cross-country skis)
To find the corner points, we need to find the intersection of these constraints.
First, let's find the intersection of the manufacturing constraints:
2x + 3y = 24
x = 0 (from the constraint x ≥ 0)
y = 8 (from substituting x = 0 into the equation)
Next, let's find the intersection of the finishing constraints:
6x + 7y = 60
x = 0 (from the constraint x ≥ 0)
y = 8.57 (from substituting x = 0 into the equation)
However, since we can't produce a fraction of a ski, we round down to the nearest whole number:
y = 8
Therefore, the corner points of the feasible region are (0, 8) and (0, 8).
Now, let's calculate the profit at each corner point:
Profit for (0, 8):
Profit = 0*71 + 8*69 = 0 + 552 = 552
Profit for (0, 8):
Profit = 0*71 + 8*69 = 0 + 552 = 552
The weekly profit is the same at both corner points, $552.
Therefore, to maximize the weekly profit, the manufacturer should make 0 downhill skis and 8 cross-country skis.
To determine the corner points of the feasible region, we need to analyze the given constraints and find the intersection points of the different lines.
Let's start by representing the constraints graphically:
Manufacturing time constraint:
2x + 3y ≤ 24
Finishing time constraint:
6x + 7y ≤ 60
Non-negativity constraint:
x ≥ 0
y ≥ 0
To find the corner points, we can set each constraint as an equation and find the intersection points:
1. Manufacturing time constraint:
2x + 3y = 24
y = (24 - 2x) / 3
2. Finishing time constraint:
6x + 7y = 60
y = (60 - 6x) / 7
Next, we can plot these equations on a graph and find the points where the lines intersect.
After plotting the lines, we can find the following intersection points:
Point A: (0, 8)
Point B: (4, 4)
Point C: (12, 0)
These intersection points represent the corners of the feasible region.
Now, we can calculate the profit at each corner point by considering the objective function:
Profit for downhill ski: $71
Profit for cross-country ski: $69
Profit at Point A: 0 downhill skis and 8 cross-country skis
Total profit = 8 x $69 = $552
Profit at Point B: 4 downhill skis and 4 cross-country skis
Total profit = (4 x $71) + (4 x $69) = $568 + $276 = $844
Profit at Point C: 12 downhill skis and 0 cross-country skis
Total profit = 12 x $71 = $852
From these calculations, we can see that Point C has the highest profit ($852).
Therefore, to maximize the weekly profit, the manufacturer should make 12 downhill skis and 0 cross-country skis.