Question
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:
$
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:
Answers
GPT 3.5
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.
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.