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 \( t \):
- Rearranging gives: \[ 36 - 12 = 6t - 2t \] \[ 24 = 4t \]
- 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 \( B(t) \):
-
Graph \( A(t) = 36 + 2t \):
Starts at \( (0, 36) \) and has a slope of \( 2 \). -
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.
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