Question
When Evelyn moved into a new house, she planted two trees in her backyard. At the time of planting, Tree A was 32 inches tall and Tree B was 16 inches tall. Each year thereafter, Tree A grew by 4 inches per year and Tree B grew by 8 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 height of both trees at the time when they have an equal height.
Answers
GPT-4o mini
To model the heights of the trees with respect to time, we can write the functions for the heights of Tree A and Tree B as follows:
- **Tree A**: The height \( AA(t) \) after \( t \) years can be represented by the equation:
\[
AA(t) = 32 + 4t
\]
where 32 is the initial height and 4 is the growth per year.
- **Tree B**: The height \( BB(t) \) after \( t \) years can be represented by the equation:
\[
BB(t) = 16 + 8t
\]
where 16 is the initial height and 8 is the growth per year.
Next, we need to find the time \( t \) when both trees have equal heights. To do this, we set the equations for the heights equal to each other:
\[
32 + 4t = 16 + 8t
\]
Now, we can solve for \( t \):
1. Subtract \( 4t \) from both sides:
\[
32 = 16 + 4t
\]
2. Subtract 16 from both sides:
\[
32 - 16 = 4t \implies 16 = 4t
\]
3. Divide both sides by 4:
\[
t = \frac{16}{4} = 4
\]
This means that both trees will be of equal height 4 years after they are planted.
Now we can find the height of both trees at that time:
- For Tree A:
\[
AA(4) = 32 + 4 \times 4 = 32 + 16 = 48 \text{ inches}
\]
- For Tree B:
\[
BB(4) = 16 + 8 \times 4 = 16 + 32 = 48 \text{ inches}
\]
At \( t = 4 \) years, both trees will be 48 inches tall.
### Graphing the Functions
1. **Tree A**:
- Height after 0 years: 32 inches
- Height after 1 year: 36 inches
- Height after 2 years: 40 inches
- Height after 3 years: 44 inches
- Height after 4 years: 48 inches
- Height after 5 years: 52 inches
2. **Tree B**:
- Height after 0 years: 16 inches
- Height after 1 year: 24 inches
- Height after 2 years: 32 inches
- Height after 3 years: 40 inches
- Height after 4 years: 48 inches
- Height after 5 years: 56 inches
You can plot these heights on the y-axis (height in inches) against time (years) on the x-axis to visually represent the growth of the trees over time. The two lines will intersect at the point (4, 48), indicating they are of equal height at that time.
- **Tree A**: The height \( AA(t) \) after \( t \) years can be represented by the equation:
\[
AA(t) = 32 + 4t
\]
where 32 is the initial height and 4 is the growth per year.
- **Tree B**: The height \( BB(t) \) after \( t \) years can be represented by the equation:
\[
BB(t) = 16 + 8t
\]
where 16 is the initial height and 8 is the growth per year.
Next, we need to find the time \( t \) when both trees have equal heights. To do this, we set the equations for the heights equal to each other:
\[
32 + 4t = 16 + 8t
\]
Now, we can solve for \( t \):
1. Subtract \( 4t \) from both sides:
\[
32 = 16 + 4t
\]
2. Subtract 16 from both sides:
\[
32 - 16 = 4t \implies 16 = 4t
\]
3. Divide both sides by 4:
\[
t = \frac{16}{4} = 4
\]
This means that both trees will be of equal height 4 years after they are planted.
Now we can find the height of both trees at that time:
- For Tree A:
\[
AA(4) = 32 + 4 \times 4 = 32 + 16 = 48 \text{ inches}
\]
- For Tree B:
\[
BB(4) = 16 + 8 \times 4 = 16 + 32 = 48 \text{ inches}
\]
At \( t = 4 \) years, both trees will be 48 inches tall.
### Graphing the Functions
1. **Tree A**:
- Height after 0 years: 32 inches
- Height after 1 year: 36 inches
- Height after 2 years: 40 inches
- Height after 3 years: 44 inches
- Height after 4 years: 48 inches
- Height after 5 years: 52 inches
2. **Tree B**:
- Height after 0 years: 16 inches
- Height after 1 year: 24 inches
- Height after 2 years: 32 inches
- Height after 3 years: 40 inches
- Height after 4 years: 48 inches
- Height after 5 years: 56 inches
You can plot these heights on the y-axis (height in inches) against time (years) on the x-axis to visually represent the growth of the trees over time. The two lines will intersect at the point (4, 48), indicating they are of equal height at that time.