Question

When Ryan moved into a new house, he planted two trees in his backyard. At the time of planting, Tree A was 16 inches tall and Tree B was 28 inches tall. Each year thereafter, Tree A grew by 7 inches per year and Tree B grew by 4 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
Attempt 1 out of 2

A, equalsA=
B, equalsB=
Tree A is taller than Tree B when tt

Answer 1

To model the heights of the trees over time, we can write the equations for the heights of Tree A and Tree B in terms of \( t \), the number of years after they were planted.

1. **For Tree A**:
- Initial height: 16 inches
- Growth rate: 7 inches per year

The equation for Tree A is:
\[
A(t) = 16 + 7t
\]

2. **For Tree B**:
- Initial height: 28 inches
- Growth rate: 4 inches per year

The equation for Tree B is:
\[
B(t) = 28 + 4t
\]

Next, we need to determine when Tree A is taller than Tree B, which can be expressed as:

\[
A(t) > B(t)
\]

Substituting the equations we derived:

\[
16 + 7t > 28 + 4t
\]

Now, let's solve this inequality:

1. Subtract \( 4t \) from both sides:
\[
16 + 3t > 28
\]

2. Subtract 16 from both sides:
\[
3t > 12
\]

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

Thus, Tree A is taller than Tree B when:

\[
t > 4
\]

This means Tree A will be taller than Tree B after 4 years of growth.

In summary:
- \( A(t) = 16 + 7t \)
- \( B(t) = 28 + 4t \)
- Tree A is taller than Tree B when \( t > 4 \).