A chemist is studying the properties of a bronze alloy (mixture) of copper and tin. She begins with 6 kg of an alloy that is one sixth tin. Keeping the amount of copper constant, she adds small amounts of tin to the alloy. Letting x be the total amount of tin added, define
c(x) = concentration of tin = Total amount of tin / total amount of alloy
find the formula for c(x)
To start with, there is 1 kg of tin.
After adding x kg of tin, then we have
c(x) = (x+1)/(x+6)
To find the formula for c(x), the concentration of tin as a function of the total amount of tin added (x), we need to consider the initial composition of the alloy and the amounts of tin added.
Given that the alloy initially contains 6 kg and is one-sixth tin, we can determine the initial amount of tin by multiplying the total amount of alloy by the fraction of tin:
Initial amount of tin = (1/6) * 6 kg = 1 kg
Now, as the chemist adds small amounts of tin (x) to the alloy, the total amount of alloy becomes (6 + x) kg, and the total amount of tin becomes (1 + x) kg.
So, the concentration of tin, c(x), is defined as the total amount of tin divided by the total amount of alloy:
c(x) = (1 + x) kg / (6 + x) kg
Therefore, the formula for c(x), the concentration of tin as a function of the total amount of tin added (x), is:
c(x) = (1 + x) / (6 + x)