Read the problem and visualize what is happening.

Question

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

Answer 1

To determine if Plant A and Plant C will ever be the same height, we can set up an equation based on their growth over time.

Let's denote:

• \( h_A(t) \): the height of Plant A after \( t \) months.
• \( h_C(t) \): the height of Plant C after \( t \) months.

Given:

• Plant A starts at 4 cm and grows at 2 cm per month.
• Plant C starts at 8 cm and also grows at 2 cm per month.

We can express the heights of the plants as functions of time:

1. For Plant A: \[ h_A(t) = 4 + 2t \]

2. For Plant C: \[ h_C(t) = 8 + 2t \]

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

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

Subtracting \( 2t \) from both sides gives:

\[ 4 = 8 \]

This is not a true statement; in fact, it shows that the equations simplify to something that indicates no solution.

This tells us that because both plants grow at the same rate (2 cm per month), the height difference between them (the 4 cm difference, where Plant C is consistently 4 cm taller) will always remain constant. Therefore, Plant A will never catch up with Plant C.

In conclusion, the two plants will never be the same height.