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When Khalil 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 16 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 interval of time, t, commat, when Tree A is taller than Tree B.

1 answer

To determine when Tree A is taller than Tree B, we first express the heights of the trees as functions of time. Let \( t \) represent the number of years since the trees were planted.

  1. Height of Tree A:

    • Initial height: 36 inches
    • Growth rate: 2 inches per year
    • Height function for Tree A, \( AA(t) \): \[ AA(t) = 36 + 2t \]

  2. Height of Tree B:

    • Initial height: 16 inches
    • Growth rate: 6 inches per year
    • Height function for Tree B, \( BB(t) \): \[ BB(t) = 16 + 6t \]

Next, we need to find the values of \( t \) for which Tree A is taller than Tree B. This condition can be expressed as: \[ AA(t) > BB(t) \] Substituting the functions: \[ 36 + 2t > 16 + 6t \]

Now, we can solve this inequality:

  1. Subtract \( 2t \) from both sides: \[ 36 > 16 + 4t \]

  2. Subtract 16 from both sides: \[ 20 > 4t \]

  3. Divide both sides by 4: \[ 5 > t \]

This can be rewritten as: \[ t < 5 \]

Thus, Tree A is taller than Tree B for \( t \) in the interval: \[ (0, 5) \]

Now let's identify the growth trajectories graphically. Here are the heights at specific years for clarity:

  • At \( t = 0 \):

    • \( AA(0) = 36 \)
    • \( BB(0) = 16 \)

  • At \( t = 1 \):

    • \( AA(1) = 38 \)
    • \( BB(1) = 22 \)

  • At \( t = 2 \):

    • \( AA(2) = 40 \)
    • \( BB(2) = 28 \)

  • At \( t = 3 \):

    • \( AA(3) = 42 \)
    • \( BB(3) = 34 \)

  • At \( t = 4 \):

    • \( AA(4) = 44 \)
    • \( BB(4) = 40 \)

  • At \( t = 5 \):

    • \( AA(5) = 46 \)
    • \( BB(5) = 46 \)

  • At \( t = 6 \):

    • \( AA(6) = 48 \)
    • \( BB(6) = 52 \)

From this data, it’s clear that Tree A remains taller than Tree B up to when \( t = 5 \) (inclusive). Up to this point, it is taller, and from \( t = 6 \) onward, Tree B becomes taller.

Final Answer

Tree A is taller than Tree B in the interval: \[ (0, 5) \]