Question

When Khalil moved into a new house, he planted two trees in his backyard. At the time of planting, Tree A was 36 inches tall and Tree B was 16 inches tall. Each year thereafter, Tree A grew by 2 inches per year and Tree B grew by 6 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. Graph each function and determine the interval of time, t, commat, when Tree A is taller than Tree B.

Answer 1

To determine when Tree A is taller than Tree B, we first express the heights of the trees as functions of time. Let \( t \) represent the number of years since the trees were planted.

1. **Height of Tree A**:
- Initial height: 36 inches
- Growth rate: 2 inches per year
- Height function for Tree A, \( AA(t) \):
\[
AA(t) = 36 + 2t
\]

2. **Height of Tree B**:
- Initial height: 16 inches
- Growth rate: 6 inches per year
- Height function for Tree B, \( BB(t) \):
\[
BB(t) = 16 + 6t
\]

Next, we need to find the values of \( t \) for which Tree A is taller than Tree B. This condition can be expressed as:
\[
AA(t) > BB(t)
\]
Substituting the functions:
\[
36 + 2t > 16 + 6t
\]

Now, we can solve this inequality:
1. Subtract \( 2t \) from both sides:
\[
36 > 16 + 4t
\]

2. Subtract 16 from both sides:
\[
20 > 4t
\]

3. Divide both sides by 4:
\[
5 > t
\]

This can be rewritten as:
\[
t < 5
\]

Thus, Tree A is taller than Tree B for \( t \) in the interval:
\[
(0, 5)
\]

Now let's identify the growth trajectories graphically. Here are the heights at specific years for clarity:
- At \( t = 0 \):
- \( AA(0) = 36 \)
- \( BB(0) = 16 \)

- At \( t = 1 \):
- \( AA(1) = 38 \)
- \( BB(1) = 22 \)

- At \( t = 2 \):
- \( AA(2) = 40 \)
- \( BB(2) = 28 \)

- At \( t = 3 \):
- \( AA(3) = 42 \)
- \( BB(3) = 34 \)

- At \( t = 4 \):
- \( AA(4) = 44 \)
- \( BB(4) = 40 \)

- At \( t = 5 \):
- \( AA(5) = 46 \)
- \( BB(5) = 46 \)

- At \( t = 6 \):
- \( AA(6) = 48 \)
- \( BB(6) = 52 \)

From this data, it’s clear that Tree A remains taller than Tree B up to when \( t = 5 \) (inclusive). Up to this point, it is taller, and from \( t = 6 \) onward, Tree B becomes taller.

### Final Answer
Tree A is taller than Tree B in the interval:
\[
(0, 5)
\]