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.
Hint:
The costs are linear, so you can either prepare it with 5%+28% or 5%+40%.
Calculate the cost of each and use the less expensive option.
I have the impression that it costs less by mixing 5% with 40%.
To find the cheapest method of preparing the 17% salt solution, we need to determine the quantities of each stock solution required while considering the cost per liter. Let's break down the process step by step:
1. Determine the amount of salt required in the 250 ml of final solution:
17% of 250 ml = 0.17 * 250 ml = 42.5 ml of salt
2. Determine the total volume of the final solution:
Since we are creating 250 ml of a solution, the total volume remains 250 ml.
3. Calculate the amount of each stock solution needed:
Let's assume we use x ml of the 5% salt, y ml of the 28% salt, and z ml of the 40% salt solution.
From the 5% salt solution:
Considering that the 5% salt solution contains 5 ml of salt per 100 ml of solution, we can set up the equation:
(5 ml / 100 ml) * x ml = 42.5 ml
Solving for x, we get:
x = (42.5 ml * 100 ml) / 5 ml = 850 ml
From the 28% salt solution:
Similarly, the 28% salt solution contains 28 ml of salt per 100 ml of solution, leading to the equation:
(28 ml / 100 ml) * y ml = 42.5 ml
Solving for y, we get:
y = (42.5 ml * 100 ml) / 28 ml ≈ 151.79 ml (rounded to two decimal places)
From the 40% salt solution:
Using the same approach, we have the equation:
(40 ml / 100 ml) * z ml = 42.5 ml
Solving for z, we get:
z = (42.5 ml * 100 ml) / 40 ml = 106.25 ml
4. Calculate the total cost of each stock solution:
To find the cost, we multiply the volume of each stock solution used by its respective cost per liter. Since we've already converted the volumes to ml, we just need to divide by 1000 to get the volumes in liters.
Cost of the 5% salt solution:
(850 ml / 1000) * $28
Cost = $23.80
Cost of the 28% salt solution:
(151.79 ml / 1000) * $38 ≈ $5.77 (rounded to two decimal places)
Cost of the 40% salt solution:
(106.25 ml / 1000) * $50 ≈ $5.31 (rounded to two decimal places)
5. Determine the total cost:
The total cost is the sum of the costs of each stock solution used:
Total Cost = Cost of 5% salt solution + Cost of 28% salt solution + Cost of 40% salt solution
≈ $23.80 + $5.77 + $5.31
≈ $34.88
Therefore, the cheapest method of preparing the 17% salt solution is by using 850 ml of the 5% salt solution, 151.79 ml of the 28% salt solution (rounded to two decimal places), and 106.25 ml of the 40% salt solution. This selection yields a total cost of approximately $34.88.