A) To write down the inequalities for x and y:
The time it takes to make the rings and necklaces should not exceed the maximum working hours of 12 hours:
2x + 3y <= 12
The goldsmith has an order for at least 2 rings and 2 necklaces:
x >= 2
y >= 2
B) To show the region that satisfies all the inequalities:
We can plot the inequalities on a graph with x as the horizontal axis and y as the vertical axis. The first inequality 2x + 3y <= 12 defines a line in the first quadrant, while the second inequalities x >= 2 and y >= 2 define a region in the first quadrant that is above the x and y axes. The region that satisfies all the inequalities is the area in the first quadrant that is below the line 2x + 3y <= 12 and above the x and y axes. This region can be shaded.
C) To maximize the profit, we need to find the x and y values that result in the greatest profit, which is equal to the number of rings multiplied by the profit per ring plus the number of necklaces multiplied by the profit per necklace. The profit equation is:
50x + 60y
Subject to the constraints:
2x + 3y <= 12
x >= 2
y >= 2
This is a linear programming problem and can be solved using optimization techniques, such as the simplex method or the graphical method.
Using the graphical method, we can find the x and y values that result in the maximum profit by finding the corner points of the region R and evaluating the profit equation at each corner point. The corner point with the highest profit is the solution to the problem.
In this case, the solution is x = 4, y = 2, which means that the goldsmith should make 4 rings and 2 necklaces to maximize the profit. The maximum profit would be $50 x 4 + $60 x 2 = $320.