Let's solve this problem step-by-step.
1. Understanding the Relationship:
- We know that the number of gallons [tex]\( g \)[/tex] is proportional to the number of seconds [tex]\( t \)[/tex]. This means [tex]\( g \)[/tex] can be expressed as a constant [tex]\( k \)[/tex] multiplied by [tex]\( t \)[/tex]:
[tex]\[
g = k \times t
\][/tex]
2. Using the Given Information:
- A 2-gallon bucket is filled in 5 seconds.
[tex]\[
g = 2 \quad \text{and} \quad t = 5
\][/tex]
3. Finding the Constant of Proportionality [tex]\( k \)[/tex]:
- We can find [tex]\( k \)[/tex] by dividing the gallons by the seconds.
[tex]\[
k = \frac{g}{t} = \frac{2}{5} = 0.4
\][/tex]
- So the relationship between [tex]\( g \)[/tex] and [tex]\( t \)[/tex] can be described by:
[tex]\[
g = 0.4 \times t
\][/tex]
4. Checking the Given Equations:
- [tex]\( g = 0.4 t \)[/tex]:
[tex]\[
g = 0.4 \times t \implies \text{This matches our derived relationship, so it's correct.}
\][/tex]
- [tex]\( t = 0.4 g \)[/tex]:
[tex]\[
t = 0.4 \times g \implies \text{This does not match as it doesn't correspond to our relationship.}
\][/tex]
- [tex]\( g = 2.5 t \)[/tex]:
[tex]\[
g = 2.5 \times t \implies \text{This does not match our derived relationship.}
\][/tex]
- [tex]\( t = 2.5 g \)[/tex]:
[tex]\[
t = 2.5 \times g \implies \text{This does not match as it doesn't correspond to our relationship.}
\][/tex]
- [tex]\( g = \frac{2}{5} t \)[/tex]:
[tex]\[
g = \frac{2}{5} t \implies \text{This matches our derived relationship.}
\][/tex]
5. Conclusion:
The equations that correctly represent the relationship between [tex]\( g \)[/tex] and [tex]\( t \)[/tex] are:
[tex]\[
g = 0.4 t \quad \text{and} \quad g = \frac{2}{5} t
\][/tex]
