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

How tall are both plants when you receive them?
Plant A is
cm tall when you receive it, and Plant B is
cm tall when you receive it.

Answer 1

When you receive the plants, their heights are as follows:

• Plant A is 4 cm tall.
• Plant B is 8 cm tall.

To determine if and when the two plants will be the same height, we can express the heights of the plants as functions of time (in months).

Let:

• \( h_A(t) \) be the height of Plant A after \( t \) months.
• \( h_B(t) \) be the height of Plant B after \( t \) months.

The heights of the plants can be expressed as:

• \( h_A(t) = 4 + 2t \) (starts at 4 cm and grows 2 cm per month)
• \( h_B(t) = 8 + 1t \) (starts at 8 cm and grows 1 cm per month)

To find out when the two plants will be the same height, we set the two equations equal to each other:

\[ 4 + 2t = 8 + 1t \]

Now, solve for \( t \):

1. Subtract \( 1t \) from both sides: \[ 4 + 2t - 1t = 8 \] \[ 4 + t = 8 \]

2. Subtract 4 from both sides: \[ t = 8 - 4 \] \[ t = 4 \]

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

Now, let's calculate their heights after 4 months:

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

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

Both plants will be 12 cm tall after 4 months.

In summary:

• Plant A is 4 cm tall when you receive it.
• Plant B is 8 cm tall when you receive it.
• The two plants will be the same height (at 12 cm) 4 months after you receive them.