Question
Plant A is 4 cm tall and grows at a constant rate of 2 cm per month. Plant B is 8 cm tall and grows at a constant rate of 1 cm per month. The relationship between time in months, t, and height in cm, h, can be represented by the equation h=4+2t for Plant A and h=8+t for Plant B.
Is there a time when the plants are the same height?
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Is there a time when the plants are the same height?
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Answers
GPT-4o mini
To find out if there is a time when the heights of Plant A and Plant B are the same, we can set their height equations equal to each other and solve for \( t \).
The equations for the heights are:
- Plant A: \( h_A = 4 + 2t \)
- Plant B: \( h_B = 8 + t \)
Setting them equal gives us:
\( 4 + 2t = 8 + t \)
Now, we can solve for \( t \):
1. Subtract \( t \) from both sides:
\( 4 + 2t - t = 8 \)
\( 4 + t = 8 \)
2. Subtract 4 from both sides:
\( t = 8 - 4 \)
\( t = 4 \)
Now let's calculate the height of both plants at \( t = 4 \):
For Plant A:
\( h_A = 4 + 2(4) = 4 + 8 = 12 \) cm
For Plant B:
\( h_B = 8 + 4 = 12 \) cm
So both plants are 12 cm tall at \( t = 4 \) months.
Therefore, the answer is **Yes**. There is a time when the plants are the same height, which occurs at \( t = 4 \) months.
The equations for the heights are:
- Plant A: \( h_A = 4 + 2t \)
- Plant B: \( h_B = 8 + t \)
Setting them equal gives us:
\( 4 + 2t = 8 + t \)
Now, we can solve for \( t \):
1. Subtract \( t \) from both sides:
\( 4 + 2t - t = 8 \)
\( 4 + t = 8 \)
2. Subtract 4 from both sides:
\( t = 8 - 4 \)
\( t = 4 \)
Now let's calculate the height of both plants at \( t = 4 \):
For Plant A:
\( h_A = 4 + 2(4) = 4 + 8 = 12 \) cm
For Plant B:
\( h_B = 8 + 4 = 12 \) cm
So both plants are 12 cm tall at \( t = 4 \) months.
Therefore, the answer is **Yes**. There is a time when the plants are the same height, which occurs at \( t = 4 \) months.