A chemist working on a flu vaccine needs to mix a 10% sodium-iodine solution with a 60% sodium-iodine solution to obtain a 50-milliliter mixture. Write the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x.
To write the amount of sodium iodine in the mixture, S, as a function of the number of milliliters of the 10% solution used, x, we need to consider the concentration of sodium iodine in each solution and the total volume of the mixture.
Let's break down the problem step by step:
1. We have a 10% sodium-iodine solution, which means it contains 10 grams of sodium iodine in every 100 milliliters of solution. Therefore, we can express this concentration mathematically as 10/100 = 0.1 g/ml.
2. Similarly, we have a 60% sodium iodine solution, which means it contains 60 grams of sodium iodine in every 100 milliliters of solution. Therefore, we can express this concentration mathematically as 60/100 = 0.6 g/ml.
3. The chemist needs to mix these two solutions to obtain a 50-milliliter mixture. Let's assume the chemist uses x milliliters of the 10% solution. Therefore, the remaining volume, (50 - x) milliliters, will come from the 60% solution.
4. The amount of sodium iodine in the mixture will be the sum of the amounts of sodium iodine from each solution. In the 10% solution, the amount will be 0.1x grams (0.1 g/ml times x ml). In the 60% solution, the amount will be 0.6(50 - x) grams (0.6 g/ml times (50 - x) ml).
5. Finally, we can express the amount of sodium iodine in the mixture, S, in milliliters, as a function of the number of milliliters of the 10% solution used, x, by converting the amount from grams to milliliters. Since the density of sodium iodine is approximately 2.28 g/ml, we can divide the amounts by 2.28 to convert them to milliliters:
S(x) = (0.1x)/2.28 + (0.6(50 - x))/2.28
Therefore, S(x) = 0.04386x + 0.26316 - 0.26316x
Simplifying the equation, we get:
S(x) = -0.2193x + 0.26316
So, the amount of sodium iodine in the mixture, S, in milliliters, is given by the function S(x) = -0.2193x + 0.26316.