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.

Question

Is there a time when the plants are the same height?
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Yes

No

Answer 1

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.