Let's say you're given two plants. When you receive Plant A, it is 4 cm tall, and it grows at a constant rate of 2 cm per month. When you receive Plant B, it is 8 cm tall, and it grows at a constant rate of 1 cm per month. Will the two plants ever be the same height? If so, when will that happen?

Question

Which plant grows at a faster rate?

Plant A

Plant B

Answer 1

To determine if both plants will ever reach the same height and when that might happen, we can set up equations to describe their growth.

1. Plant A:

   • Initial height: 4 cm
   • Growth rate: 2 cm per month
   • Height after \( t \) months: \[ H_A(t) = 4 + 2t \]

2. Plant B:

   • Initial height: 8 cm
   • Growth rate: 1 cm per month
   • Height after \( t \) months: \[ H_B(t) = 8 + 1t \]

Next, we can set the heights of both plants equal to each other to find out when they will be the same height: \[ 4 + 2t = 8 + 1t \]

Now, solve for \( t \): \[ 2t - 1t = 8 - 4 \] \[ t = 4 \]

So, the two plants will be the same height after 4 months.

Now let's calculate their heights at that time:

• For Plant A: \[ H_A(4) = 4 + 2(4) = 4 + 8 = 12 \text{ cm} \]

• For Plant B: \[ H_B(4) = 8 + 1(4) = 8 + 4 = 12 \text{ cm} \]

Thus, both plants will indeed reach 12 cm in height after 4 months.

Finally, to answer which plant grows at a faster rate, we can simply compare their growth rates:

• Plant A grows at 2 cm per month.
• Plant B grows at 1 cm per month.

Therefore, Plant A grows at a faster rate.