You are a lab technician and must create 250 ml of a 17% salt solution. You have available three stock solutions. You have a one liter container of a 5% salt, a 500 ml container of a 28% salt solution, and a 400 ml container of a 40% salt solution. Show the work necessary to calculate the cheapest method of preparing the 17% salt solution if the 5% salt solution costs $28 per liter, the 28% solution costs $38 per liter, and the 40% solution costs $50 per liter. Be sure to explain in paragraph form why you have selected the amounts of each and the total cost of your selection.
let c = cost of final solution
use x,y,z liters of the three solutions
x <= 1.0
y <= .5
z <= .4
x + y + z = .25
.05x + .28y + .40z = .17*.25
c = 28x + 38y + 50z
A little manipulation of tableaux (a google on linear programming will turn up zweigmedia) will show that
Optimal Solution: c = 8.89; x = 0.164, y = 0, z = 0.086
Hmmm. no 28% solution used.
??I got this when I followed steve's steps...
optimal solution: c=191/23; x=2750/23; y=3000/23; z=0
my x and y are in ml not liters..
To calculate the cheapest method of preparing a 17% salt solution, we need to determine the most cost-effective combination of the available stock solutions. Let's break down the problem step by step.
First, we need to determine the final volume of the 17% salt solution, which is 250 ml or 0.25 liters.
Next, let's consider the percentage of salt in each stock solution and their respective costs per liter:
1. The 5% salt solution costs $28 per liter.
2. The 28% salt solution costs $38 per liter.
3. The 40% salt solution costs $50 per liter.
To simplify the calculations, we can convert the costs to cents per milliliter:
1. The 5% salt solution costs 28 cents per milliliter.
2. The 28% salt solution costs 38 cents per milliliter.
3. The 40% salt solution costs 50 cents per milliliter.
To determine the cheapest method, we can compare the costs of preparing the 17% salt solution using different combinations.
Case 1: Using only the 5% salt solution:
To achieve a 17% salt solution, we can calculate the amount needed using the formula:
(5% concentration) x (volume) = (17% concentration) x (volume)
(0.05) x (volume) = (0.17) x (0.25)
volume = (0.17 x 0.25) / 0.05
volume = 0.85 liters
The cost of using only the 5% salt solution would be:
Cost = (0.85 liters) x ($28 per liter)
Cost = $23.80
Case 2: Using only the 28% salt solution:
Using the same calculation, we find that we would need:
volume = (0.17 x 0.25) / 0.28
volume = 0.1518 liters
The cost of using only the 28% salt solution would be:
Cost = (0.1518 liters) x ($38 per liter)
Cost = $5.77
Case 3: Using only the 40% salt solution:
Similarly, we find that we would need:
volume = (0.17 x 0.25) / 0.40
volume = 0.10625 liters
The cost of using only the 40% salt solution would be:
Cost = (0.10625 liters) x ($50 per liter)
Cost = $5.31
Case 4: Combining the 5% and 28% salt solutions:
To find the optimal combination at the lowest cost, we need to compare the cost of this case with the previous cases.
First, we calculate the volume of each solution needed:
Volume of 5% solution = (0.05) x (volume) = (0.05) x (0.25) = 0.0125 liters
Volume of 28% solution = (0.17 - 0.05) x (volume) / (0.28 - 0.05) = 0.1176 liters
The total cost of combining the 5% and 28% salt solutions would be:
Cost = (0.0125 liters) x ($28 per liter) + (0.1176 liters) x ($38 per liter)
Cost = $5.36
Case 5: Combining the 5% and 40% salt solutions:
Using the same approach, we find:
Volume of 5% solution: 0.035 liters
Volume of 40% solution: 0.085 liters
Total cost:
Cost = (0.035 liters) x ($28 per liter) + (0.085 liters) x ($50 per liter)
Cost = $6.40
Case 6: Combining the 28% and 40% salt solutions:
Similarly:
Volume of 28% solution: 0.2052 liters
Volume of 40% solution: 0.0448 liters
Total cost:
Cost = (0.2052 liters) x ($38 per liter) + (0.0448 liters) x ($50 per liter)
Cost = $10.12
After evaluating all these cases, we find that the cheapest method is Case 3, using only the 40% salt solution. This method costs $5.31, which provides the most cost-effective way to prepare a 250 ml 17% salt solution.