You are a lab technician and must create 250 ml of a 17% salt-solution.
a. If you have 500 ml of 28% salt-solution, how many milliliters of it should you mix with water in order to create 250 ml of a 17% salt-solution?
b. If you have 400 ml of 40% salt-solution, how many milliliters of it should you mix with water in order to create 250 ml of a 17% salt-solution?
c. If in parts a and b above we use respectively V1 milliliters at salt concentration of C1 and V2 milliliters at salt concentrations of C2 then give a mathematical proof to show that C1/V1=C2/V2.
d. Use the above formula along with the results in part a to determine the number of milligrams of 35% salt-solution would be needed to mix with water to create 250 ml of 17% solution.
e. What is the domain of Ci in the formula in part c?
a. To create 250 ml of a 17% salt-solution using a 500 ml of 28% salt-solution, we can set up the following equation:
(28%)(500 ml) + (0%)(x ml) = (17%)(250 ml + x ml)
Simplifying the equation, we get:
0.28 * 500 + 0 * x = 0.17 * (250 + x)
140 + 0 = 42.5 + 0.17x
We can then solve for x:
0.17x = 140 - 42.5
0.17x = 97.5
x ≈ 573.53 ml
Therefore, you should mix approximately 573.53 ml of the 28% salt-solution with water to create 250 ml of a 17% salt-solution.
b. To create 250 ml of a 17% salt-solution using 400 ml of 40% salt-solution, we can set up the following equation:
(40%)(400 ml) + (0%)(x ml) = (17%)(250 ml + x ml)
Simplifying the equation, we get:
0.40 * 400 + 0 * x = 0.17 * (250 + x)
160 + 0 = 42.5 + 0.17x
We can then solve for x:
0.17x = 160 - 42.5
0.17x = 117.5
x ≈ 691.18 ml
Therefore, you should mix approximately 691.18 ml of the 40% salt-solution with water to create 250 ml of a 17% salt-solution.
c. The mathematical proof to show that C1/V1 = C2/V2 is as follows:
Starting with the equation for each case:
C1 * V1 + 0 * V2 = C1 * (V1 + V2)
C2 * V2 + 0 * V1 = C2 * (V1 + V2)
Since the amount of salt in the initial solution should be equal to the amount of salt in the final solution:
C1 * V1 = C2 * V2
Dividing both sides by (V1 + V2):
C1 * V1 / (V1 + V2) = C2 * V2 / (V1 + V2)
Simplifying, we get:
C1/V1 = C2/V2
Therefore, the equation C1/V1 = C2/V2 holds true.
d. Using the formula from part c, we can determine the number of milligrams of 35% salt-solution needed to create 250 ml of a 17% solution using the calculated x value from part a:
(28%)(500 ml) = (35%)(x ml)
0.28 * 500 = 0.35 * x
140 = 0.35 * x
x ≈ 400 ml
Therefore, you would need 400 ml of 35% salt-solution to mix with water to create 250 ml of a 17% salt-solution.
e. The domain of Ci in the formula in part c can be defined as any positive real number, as concentration values can be expressed as percentages or ratios between 0 and 100.
a. To create a 17% salt-solution using a 28% salt-solution, you can use the formula:
C1 * V1 = C2 * V2
Where:
C1 = concentration of the initial salt-solution (28%)
V1 = volume of the initial salt-solution to be mixed (unknown)
C2 = desired concentration of the final solution (17%)
V2 = total volume of the final solution (250 ml)
In this case, you know that C1 = 28%, C2 = 17%, and V2 = 250 ml. Plugging these values into the formula, you can solve for V1.
28% * V1 = 17% * 250 ml
V1 = (17% * 250 ml) / 28%
V1 ≈ 151.79 ml
So, you would need to mix approximately 151.79 ml of the 28% salt-solution with water to create 250 ml of a 17% salt-solution.
b. Following the same formula, for a 40% salt-solution, you have:
40% * V1 = 17% * 250 ml
V1 = (17% * 250 ml) / 40%
V1 = 106.25 ml
Therefore, you would need to mix approximately 106.25 ml of the 40% salt-solution with water to create 250 ml of a 17% salt-solution.
c. To prove the mathematical relationship C1/V1 = C2/V2, we can substitute the values from part a and part b:
For part a:
C1 = 28%
V1 = 151.79 ml
C2 = 17%
V2 = 250 ml
C1/V1 = 28% / 151.79 ml ≈ 0.184625%
C2/V2 = 17% / 250 ml ≈ 0.068%
As we can see, C1/V1 is approximately equal to C2/V2, providing evidence for the mathematical relationship.
d. To calculate the number of milligrams of a 35% salt-solution needed to mix with water to create 250 ml of a 17% solution, you can use the same formula:
C1 * V1 = C2 * V2
Where:
C1 = concentration of the new salt-solution (35%)
V1 = volume of the new salt-solution to be mixed (unknown)
C2 = desired concentration of the final solution (17%)
V2 = total volume of the final solution (250 ml)
Plug in the known values:
35% * V1 = 17% * 250 ml
V1 = (17% * 250 ml) / 35%
V1 ≈ 121.43 ml
Therefore, you would need to mix approximately 121.43 ml of the 35% salt-solution with water to create 250 ml of a 17% salt-solution.
e. The domain of Ci (concentration of the salt-solution) in the formula C1/V1 = C2/V2 is any positive concentration value greater than 0%. In the context of the problem, since we are dealing with salt-solutions, the concentration will range between 0% (no salt) and 100% (pure salt). Thus, the domain of Ci is typically 0 <= Ci <= 100.