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.
How can you use the graph to determine that the plants will never be the same height?
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The lines are
so there
.

Answer 1

To determine whether Plant A and Plant C will ever be the same height using a graph, you can plot their growth over time on a coordinate system where the x-axis represents time (in months) and the y-axis represents height (in centimeters).

1. **Define Height as a Function of Time**:
- For Plant A: The height \( h_A \) as a function of time \( t \) can be expressed as:
\[
h_A = 4 + 2t
\]
- For Plant C: The height \( h_C \) as a function of time \( t \) can be expressed as:
\[
h_C = 8 + 2t
\]

2. **Plot the Growth on the Graph**:
- Start with Plant A, which is initially at 4 cm (when \( t = 0 \)) and grows at a rate of 2 cm per month. The line for Plant A starts at (0, 4) and has a slope of 2.
- For Plant C, start at 8 cm (when \( t = 0 \)) and also grows at a rate of 2 cm per month. The line for Plant C starts at (0, 8) and also has a slope of 2.

3. **Analyzing the Lines**:
- Both plants have the same growth rate (2 cm per month), which means that their rate of increase in height is identical. This results in parallel lines on the graph.
- Since Plant A starts at 4 cm and Plant C starts at 8 cm, the distance between the two lines remains constant as they grow.

4. **Conclusion**:
- Because the lines are parallel, they will never intersect, which means that the heights of Plant A and Plant C will never be the same, regardless of how much time passes.

So, to summarize:
The lines are **parallel**, so there **will never be a point where the heights of Plant A and Plant C are the same**.