To determine the total capacity of the container, let's go through the given information step by step.
1. Initial Information:
- The container is initially [tex]\( \frac{1}{5} \)[/tex] full.
- After pouring in 14 gallons of water, the container is [tex]\( \frac{7}{10} \)[/tex] full.
2. Identify the change in the container's fullness:
- Initial proportion: [tex]\( \frac{1}{5} = 0.2 \)[/tex]
- Final proportion: [tex]\( \frac{7}{10} = 0.7 \)[/tex]
The change in the proportion of the container's capacity (fullness) is:
[tex]\[
0.7 - 0.2 = 0.5
\][/tex]
3. Relate the change in fullness to the added volume:
- The change in the proportion of the container's capacity is 0.5.
- This change corresponds to the 14 gallons of water that were poured in.
4. Calculate the total capacity of the container:
- If 0.5 of the container’s capacity equals 14 gallons, we can set up the proportion to find the total capacity [tex]\( C \)[/tex]:
[tex]\[
0.5C = 14
\][/tex]
- Solving for [tex]\( C \)[/tex]:
[tex]\[
C = \frac{14}{0.5} = 28
\][/tex]
Therefore, the total capacity of the container is:
[tex]\[
\boxed{28}
\][/tex]
gallons.
