Question

When Jocelyn moved into a new house, she planted two trees in her backyard. At the time of planting, Tree A was 25 inches tall and Tree B was 37 inches tall. Each year thereafter, Tree A grew by 10 inches per year and Tree B grew by 7 inches per year. Let AA represent the height of Tree A tt years after being planted and let BB represent the height of Tree B tt years after being planted. Write an equation for each situation, in terms of t, commat, and determine the interval of time, t, commat, when Tree A is taller than Tree B.

Answer 1

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 \( t > 4 \).