Answer:
Variables
X = number of pots made and Y = number of plates made
Constraints
2X + 1Y ≤ 50
X ≤ 24
Y ≤ 40
X ≥ 0, Y ≥ 0
Objective Equation
P = 25X + 12Y
where P is profit in $
Solution:
24 pots and 2 plates for a total profit of $624
Explanation:
This is an example of a Linear Programming( LP) problem where the objective is to maximize or minimize a function subject to constraints (inequalities) which are linear in nature
To proceed, first define variables to represent the number of pots and pans Savannah has to make
Let X represent the number of pots and Y the number of plates that Savannah finally makes at maximum profit
Constraint 1: Weight constraint
Since each pot weighs 2 pounds, X pots will weigh 2X pounds
Each plate weighs 1 pound so Y plates will weigh 1Y pounds
The sum of these weights cannot exceed 50 pounds i.e ≤ 50 pounds
So first constraint is
2X + 1Y ≤ 50
Constraint 2: Clay resource constraint
Savannah has only enough clay to make 40 plates . So the number of plates she can make is subject to the constraint
Y ≤ 40
Similarly with only enough clay to make 24 pots, the next resource constraint would be
X ≤ 24
(Remember X = number of pots and Y = number of plates - they are trying to trick you by interchanging pots and pans in the statements)
Each pot gets her a profit of $25 and each plate gives her a profit of $12
So the profit P she can expect to get by selling X pots and Y plates is given by the equation
P = 25X + 12Y
This is the objective function or objective equation
So here are the answers
Variables: X = number of pots made and Y = number of plates made
Constraints
2X + 1Y ≤ 50 (Weight Constraint)
X ≤ 24 (Clay constraint for pots)
Y ≤ 40 ((Clay constraint for pots)
X ≥ 0, Y ≥ 0 ( we add these to make sure X, Y cannot be negative)
Objective Equation : P = 25X + 12Y
Objective is to maximize 25X + 12Y
The entire LP problem is stated as below
Max 25X + 12Y
subject to:
2X + 1Y ≤ 50 (Weight Constraint)
X ≤ 24 (Clay constraint for pots)
Y ≤ 40 ((Clay constraint for pots)
X ≥ 0, Y ≥ 0
To solve:
Best solved using an LP calculator but I will provide the graphical solution method
If we graph all these inequalities we get a "feasible" region which is the region falling within all inequalities. We select the corner points of this region and plug in these X and Y values inside the objective equation and see which (X,Y) maximizes
Look at the attached graph:
The feasible region is the figure ABCD with corner points
A(0,0), B(0, 40), C(5,40), D(24, 2) and E (24,2)
The maximum will correspond to one of these corner points
We can easily see that C(5,40) is superior to B(0,40) so discard B as a possible solution point
Similarly we can see that D(24,2) is superior to D(24,0) so discard point E
We have two choices corresponding to points C and D. Plugging in the
X, Y values of these two points in the objective equation will give us the value of profit at each of those points
Choose the X, Y values which have a higher P value
At C(5, 40) we get P = 25 x 5 + 12 x 40 = $605
At D(24, 2) we get P = 25 x 24 + 2 x 12 = $624
Therefore Savannah should make 24 pots and 2 plates for a total profit of $624