Let's solve the given problem step-by-step to find the volume of the 35% acetic acid solution needed to create a 15% acetic acid mixture when mixed with 1 gallon of 10% acetic acid solution.
### Information Given:
1. 10% Acetic Acid Solution:
- Volume = 1 gallon
- Concentration = 10% (or 0.10)
2. 35% Acetic Acid Solution:
- Volume = [tex]\(v\)[/tex] (unknown, what we need to find)
- Concentration = 35% (or 0.35)
3. Final Mixture:
- Concentration = 15% (or 0.15)
### Steps to Solve the Problem:
#### Step 1: Calculate the Amount of Acid in the 10% Solution
[tex]\[ \text{Amount of acid in 10% solution} = \text{Volume} \times \text{Concentration} \][/tex]
[tex]\[ \text{Amount of acid in 10% solution} = 1 \times 0.10 = 0.10 \text{ gallons} \][/tex]
#### Step 2: Express the Amount of Acid in the 35% Solution
[tex]\[ \text{Amount of acid in 35% solution} = v \times 0.35 \][/tex]
(where [tex]\(v\)[/tex] is the volume in gallons that we need to determine)
#### Step 3: Express the Total Volume and Concentration of the Mixture
Total volume of the mixture:
[tex]\[ \text{Total volume} = 1 + v \][/tex]
Amount of acid in the final mixture:
[tex]\[ \text{Amount of acid in the mixture} = \text{Total volume} \times \text{Final concentration} \][/tex]
[tex]\[ \text{Amount of acid in the mixture} = (1 + v) \times 0.15 \][/tex]
#### Step 4: Set Up the Equation
The sum of the acids in the individual solutions in the mixture should equal the amount of acid in the final mixture:
[tex]\[ \text{Amount of acid in 10% solution} + \text{Amount of acid in 35% solution} = \text{Amount of acid in the final mixture} \][/tex]
Substitute in the known values:
[tex]\[ 0.10 + 0.35v = (1 + v) \times 0.15 \][/tex]
#### Step 5: Simplify the Equation
First, distribute the [tex]\(0.15\)[/tex] on the right-hand side:
[tex]\[ 0.10 + 0.35v = 0.15 + 0.15v \][/tex]
Move all terms involving [tex]\(v\)[/tex] to one side and constant terms to the other side:
[tex]\[ 0.35v - 0.15v = 0.15 - 0.10 \][/tex]
[tex]\[ 0.20v = 0.05 \][/tex]
Solve for [tex]\(v\)[/tex]:
[tex]\[ v = \frac{0.05}{0.20} \][/tex]
[tex]\[ v = 0.25 \][/tex]
### Solution:
The volume of the 35% acetic acid solution needed is 0.25 gallons.
