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:
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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.
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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 \( t \):
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Subtract \( 4t \) from both sides: \[ 32 = 16 + 4t \]
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Subtract 16 from both sides: \[ 32 - 16 = 4t \implies 16 = 4t \]
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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:
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For Tree A: \[ AA(4) = 32 + 4 \times 4 = 32 + 16 = 48 \text{ inches} \]
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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
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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
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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.
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