A manufacturer has 600 litres of a 12% solution of acid. How many litres of a 30% acid solution must be added to it so that acid content in the resulting mixture will be more than 15% but less than 18% ?
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Step-by-Step Solution
Step 1: Define variables and initial acid content
Let x be the volume (in litres) of the 30% acid solution to be added. The initial solution has a volume of 600 litres and contains 12% acid. We need to find the range of x such that the final mixture's acid concentration is between 15% and 18%.
Step 2: Calculate total acid and total volume in the mixture
First, calculate the amount of acid in the initial 600 litres of 12% solution. Then, express the amount of acid in the x litres of 30% solution. The total acid in the mixture will be the sum of these two amounts, and the total volume will be the sum of the initial volume and x.
Step 3: Set up inequalities for the acid concentration
The problem states that the acid content in the resulting mixture must be more than 15% but less than 18%. This can be written as an inequality where the ratio of total acid to total volume is between 0.15 and 0.18.
Step 4: Solve the first inequality
We solve the first part of the inequality: the acid concentration must be greater than 15%. Multiply both sides by (600+x), distribute, and then isolate x to find its lower bound.
Step 5: Solve the second inequality
Next, we solve the second part of the inequality: the acid concentration must be less than 18%. Similar to the previous step, multiply by (600+x), distribute, and isolate x to find its upper bound.
Step 6: Combine the inequalities
By combining the results from both inequalities, we find the range of x for which the acid content in the resulting mixture will be more than 15% but less than 18%.