Sure, let's go through the problem step-by-step to find the value of [tex]\( x \)[/tex], the acid concentration of the first solution.
1. Given Information:
- We have 4 liters of one acid solution with an unknown concentration [tex]\( x \)[/tex].
- We have 10 liters of a 40% acid solution.
- The final mixture is 14 liters with a 30% acid concentration.
2. Set Up the Equation:
To find [tex]\( x \)[/tex], we will set up an equation based on the amount of pure acid in each solution.
The amount of pure acid in the first solution is [tex]\( 4 \times x \)[/tex].
The amount of pure acid in the 40% solution is [tex]\( 10 \times 0.40 = 4 \)[/tex] liters.
The amount of pure acid in the final mixture is [tex]\( 14 \times 0.30 = 4.2 \)[/tex] liters.
3. Form the Equation:
We know that the total amount of acid in the mixture is the sum of the amounts of pure acid from each solution. So, we set up the following equation:
[tex]\[
4x + 4 = 4.2
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
[tex]\[
4x + 4 = 4.2
\][/tex]
Subtract 4 from both sides:
[tex]\[
4x = 0.2
\][/tex]
Divide by 4:
[tex]\[
x = \frac{0.2}{4} = 0.05
\][/tex]
So, the concentration of the first solution is [tex]\( 5\% \)[/tex].
Thus, the value of [tex]\( x \)[/tex] is [tex]\( \boxed{0.05} \)[/tex].
