A chain saw requires 5 hours of assembly and a wood chipper 6 hours a maximum of 60 hours of assembly time is available the profit is $190 on a chain saw and $220 on a chipper how many of each should be assembled for maximum profit
you want to maximize p = 190x+220y subject to the constraints
x>=0
y>=0
5x+6y <= 60
so evaluate p at (0,10), (12,0) to find the maximum
p = 190 s + 220 c
60 >/= 5 s + 6 c
s on x axis , c on y axis
5 s + 6 c </= 60 is left of line from point A = (0,10) down to Point B (12,0)
check p along that line { c = 10 - (10/12) s }
at A
p = 190 *0 + 220 * 10 = 2200
at B
p =190 *12 + 220 *0 = 2280
try elsewhere on the line, for example ( 6, 5)
p = 190 * 6 + 220 * 5 = 1140 + 1100= 2240 at point B
in fact I doubt you will find any point on that line with profit more than2280
at Point B
To determine how many of each item should be assembled for maximum profit, we can use linear programming.
Let's define our variables:
Let x be the number of chain saws to be assembled.
Let y be the number of wood chippers to be assembled.
Now, we can set up the constraints based on the given information:
Constraint 1: Assembly time constraint
5x + 6y ≤ 60
This constraint ensures that the total assembly time does not exceed the maximum available time of 60 hours.
Constraint 2: Non-negativity constraint
x ≥ 0 and y ≥ 0
This constraint ensures that the number of each item is non-negative.
To maximize profit, we need to define the objective function:
Objective function: Profit
Z = 190x + 220y
Now, we can solve this linear programming problem using techniques such as graphical method, simplex method, or an online solver.
Let's use the graphical method to solve this problem:
1. Graph the feasible region:
To graph the feasible region, we can graph the lines corresponding to the constraints and shade the region that satisfies all the constraints.
For the constraint 5x + 6y ≤ 60:
Plot the line 5x + 6y = 60 by finding its x-intercept and y-intercept. Connect these two points to draw the line.
For the non-negativity constraint, x ≥ 0 and y ≥ 0:
Plot x = 0 (y-axis) and y = 0 (x-axis) as vertical and horizontal lines.
2. Determine the corner points of the feasible region:
The corner points of the feasible region are the vertices of the shaded region.
3. Evaluate the objective function at each corner point:
Calculate the profit (Z) using the objective function for each corner point.
4. Identify the point that maximizes profit:
Identify the corner point with the highest profit. The corresponding values of x and y will give you the quantity of chain saws and wood chippers to be assembled for maximum profit.
Once you have solved the problem using the method of your choice, you will have the optimal quantities of chain saws (x) and wood chippers (y) to assemble for maximum profit.