Question 938303: You have 180 tomatoes and 15 onions left over from your garden. You want to use these to make jars of tomato sauce and jars of salsa to sell at a farm stand. A jar of tomato sauce requires 10 tomatoes and 1 onion, and a jar of salsa requires 5 tomatoes and 1/4 onion. You'll make a profit of $2 on every tomato sauce jar sold and a profit of $1.50 on every jar of salsa sold. The farm stand wants at least 3 times as many jars of tomato sauce as jars of salsa. How many jars of each should you make to maximize profit?
Answer by mathmate(429)   (Show Source):
You can put this solution on YOUR website! Linear-systems/938303 (2015-01-12 22:20:50): You have 180 tomatoes and 15 onions left over from your garden. You want to use these to make jars of tomato sauce and jars of salsa to sell at a farm stand. A jar of tomato sauce requires 10 tomatoes and 1 onion, and a jar of salsa requires 5 tomatoes and 1/4 onion. You'll make a profit of $2 on every tomato sauce jar sold and a profit of $1.50 on every jar of salsa sold. The farm stand wants at least 3 times as many jars of tomato sauce as jars of salsa. How many jars of each should you make to maximize profit?
Given:
180 tomatoes
15 onions
For each jar of sauce: 10 tomatoes and 1 onion
For each jar of salsa: 5 tomatoes and 1/4 onion.
Profit for sauce: $2 per jar
Profit for salsa: $1.50 per jar
Set up constraints:
Let number of jars of sauce = x (horizontal axis)
Let number of jars of salsa = y (vertical axis)
number of tomatoes required : 10x + 5y <= 180....(1)
number of onions required : x + y/4 <=15....(2)
#jars of sauce at least equals 3 times #jars of salsa : x >= 3y....(3)
x>=0....(4)
y>=0....(5)
x,y (number of jars) must be integers!
Note:
<= is less than or equal to
>= is greater than or equal to
Objective function, Z:
Let objective function Z
= Total profit, Z(x,y) = 2x+1.5y
Graphing.
Graphing is not required to solve the problem, but it gives a visual interpretation of the problem and the solution, and is highly recommended.
Since most of us like to work with integers and simplified equations, we will simplify inequalities (1) and (2) as follows:
2x+y <= 36......(1a)
4x+y <= 60......(2a)
x >= 3y.....(3a)
x>=0......(4a)
y>=0......(5a)
Study the graph shown below that illustrates the solution:
The shaded feasible region is enclosed by three oblique lines intersecting at D(13.85,4.62) and the two axes.
One of the family of lines of the optimal objective function Z(x,y)=2x+1.5y is shown passing through point D. This line has a constant slope and its value (profit) increases as it gets higher in the graph, but all lines of the family must touch at least one point in the feasible region for it to be valid.
Since we cannot make fractional number of jars, we have to reduce the values of x and y to integers.
We can do this by plotting integral points around point D, and working only with those points which are integers. Two such candidates are J(14,4) and K(13,4). Point L is rejected because it lies outside the feasible region.
By sliding the Z(x,y) line through J or K, it is clear that J(14,4) will give a higher profit, but we can still double check:
For J(14,4), profit = 2(14)+1.5(4) = $34
For K(13,4), profit = 2(13)+1.5(4) = $32 (smaller profit than point J)
Therefore we opt for making 14 jars of sauce, and 4 jars of salsa.
I leave you with the finishing touch of double-checking that all five constraints are satisfied.
Conclusion
The maximum profit is at Z(14,4)=2(14)+1.5(4) = $34.
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