Question
When Jordan 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 12 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 number of years after the trees were planted when both trees have an equal height.
Answers
GPT-4o mini
To model the heights of the trees over time, we can create equations based on the growth patterns described.
Let \( t \) represent the number of years after planting.
**For Tree A**:
- Initial height: 36 inches
- Growth rate: 2 inches per year
The height \( A(t) \) of Tree A after \( t \) years is given by:
\[
A(t) = 36 + 2t
\]
**For Tree B**:
- Initial height: 12 inches
- Growth rate: 6 inches per year
The height \( B(t) \) of Tree B after \( t \) years is given by:
\[
B(t) = 12 + 6t
\]
Next, we want to find out when both trees have the same height. To do this, we set the two height equations equal to each other:
\[
36 + 2t = 12 + 6t
\]
Now, let's solve for \( t \):
1. Rearranging gives:
\[
36 - 12 = 6t - 2t
\]
\[
24 = 4t
\]
2. Dividing both sides by 4:
\[
t = 6
\]
Thus, both trees will be the same height after 6 years.
**Calculating the height of both trees at \( t = 6 \)**:
- For Tree A:
\[
A(6) = 36 + 2 \cdot 6 = 36 + 12 = 48 \, \text{inches}
\]
- For Tree B:
\[
B(6) = 12 + 6 \cdot 6 = 12 + 36 = 48 \, \text{inches}
\]
Both trees will be 48 inches tall after 6 years.
**Summary**:
- The trees will be the same height \( 6 \) years after being planted.
- At that time, both will be \( 48 \) inches tall.
To graph the functions \( A(t) \) and \( B(t) \):
1. **Graph \( A(t) = 36 + 2t \)**:
Starts at \( (0, 36) \) and has a slope of \( 2 \).
2. **Graph \( B(t) = 12 + 6t \)**:
Starts at \( (0, 12) \) and has a slope of \( 6 \).
You will find that both lines will intersect at the point \( (6, 48) \) on the graph, confirming that they are equal in height at that point in time.
Let \( t \) represent the number of years after planting.
**For Tree A**:
- Initial height: 36 inches
- Growth rate: 2 inches per year
The height \( A(t) \) of Tree A after \( t \) years is given by:
\[
A(t) = 36 + 2t
\]
**For Tree B**:
- Initial height: 12 inches
- Growth rate: 6 inches per year
The height \( B(t) \) of Tree B after \( t \) years is given by:
\[
B(t) = 12 + 6t
\]
Next, we want to find out when both trees have the same height. To do this, we set the two height equations equal to each other:
\[
36 + 2t = 12 + 6t
\]
Now, let's solve for \( t \):
1. Rearranging gives:
\[
36 - 12 = 6t - 2t
\]
\[
24 = 4t
\]
2. Dividing both sides by 4:
\[
t = 6
\]
Thus, both trees will be the same height after 6 years.
**Calculating the height of both trees at \( t = 6 \)**:
- For Tree A:
\[
A(6) = 36 + 2 \cdot 6 = 36 + 12 = 48 \, \text{inches}
\]
- For Tree B:
\[
B(6) = 12 + 6 \cdot 6 = 12 + 36 = 48 \, \text{inches}
\]
Both trees will be 48 inches tall after 6 years.
**Summary**:
- The trees will be the same height \( 6 \) years after being planted.
- At that time, both will be \( 48 \) inches tall.
To graph the functions \( A(t) \) and \( B(t) \):
1. **Graph \( A(t) = 36 + 2t \)**:
Starts at \( (0, 36) \) and has a slope of \( 2 \).
2. **Graph \( B(t) = 12 + 6t \)**:
Starts at \( (0, 12) \) and has a slope of \( 6 \).
You will find that both lines will intersect at the point \( (6, 48) \) on the graph, confirming that they are equal in height at that point in time.