To establish the equations for the heights of Tree A and Tree B as functions of time \( t \) (in years), we can start by incorporating the initial heights and the growth rates:
1. **Height of Tree A after t years**:
\[
A(t) = 25 + 10t
\]
This equation indicates that Tree A starts at 25 inches and grows 10 inches each year.
2. **Height of Tree B after t years**:
\[
B(t) = 37 + 7t
\]
This expresses that Tree B starts at 37 inches and grows 7 inches each year.
Next, we want to find the interval of time \( t \) during which Tree A is taller than Tree B. This occurs when:
\[
A(t) > B(t)
\]
Substituting the height equations:
\[
25 + 10t > 37 + 7t
\]
Now let's solve this inequality step by step:
1. **Subtract \( 7t \) from both sides**:
\[
25 + 3t > 37
\]
2. **Subtract 25 from both sides**:
\[
3t > 12
\]
3. **Divide by 3**:
\[
t > 4
\]
Thus, Tree A will be taller than Tree B for any time \( t \) greater than 4 years. This means:
### Interval
\[
t > 4
\]
In interval notation, this is expressed as:
\[
(4, \infty)
\]
To summarize:
- The height equations are \( A(t) = 25 + 10t \) and \( B(t) = 37 + 7t \).
- Tree A is taller than Tree B for \( t > 4 \).