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

Are both plants the same height when you receive them?

Yes

No

Answer 1

To determine if the two plants will ever be the same height, we can set up equations for their heights over time.

Let's denote the number of months after receiving the plants as \( t \).

• The height of Plant A after \( t \) months: \[ \text{Height of Plant A} = 4 + 2t \]

• The height of Plant B after \( t \) months: \[ \text{Height of Plant B} = 8 + 1t \]

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, we solve for \( t \):

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

2. Subtract 4 from both sides: \[ 1t = 8 - 4 \] Simplifying gives: \[ 1t = 4 \]

3. Thus, \( t = 4 \).

Now, we can check their heights at \( t = 4 \):

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

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

Both plants will be the same height of 12 cm after 4 months.

Now, to answer your final question, are both plants the same height when you receive them?

• Plant A is 4 cm and Plant B is 8 cm, so No, they are not the same height when you receive them.

In summary:

• Yes, the plants will be the same height after 4 months.
• No, they are not the same height when you receive them.