https://file.aiquickdraw.com/mathbot/file/1726792371984ihv7606m.jpg

Let's analyze the question and the graph presented.

The graph shows the height \( h \) (in inches) of a bamboo plant over time, measured in months \( t \). 

### Part a: What is the slope of this line?
To find the slope \( m \), we use the formula:
\[
m = \frac{\Delta h}{\Delta t} = \frac{\text{change in height}}{\text{change in time}}
\]
From the graph, it shows that the height increases by 15 inches (from 15 to 30 inches) over a span of 5 months (from 0 to 5 months). So, the slope is:
\[
m = \frac{15}{5} = 3
\]
Thus, the slope of the line is \( 3 \).

### Part b: What does that value mean in this context?
The slope of 3 means that the bamboo plant grows by 3 inches every month. This is the rate of growth of the bamboo.

### Part c: At what point does the line intersect the h-axis?
The line intersects the \( h \)-axis when \( h = 0 \). In this case, the graph starts at \( t = 0 \) and \( h = 15 \). This means the line does not intersect the \( h \)-axis at any positive value of \( t \); rather, it intersects at a point corresponding to when the plant's height was 15 inches at the time of planting.

### Part d: What does that value mean in this context?
The intercept of \( h = 15 \) means that the bamboo plant was already 15 inches tall at the time it was planted (i.e., at \( t = 0 \)).

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Would you like more detailed explanations on any part of this solution?

Here are some related questions:
1. How would the graph change if the bamboo grew faster?
2. What happens to the graph if the initial height is different?
3. How does the slope affect the overall shape of the line?
4. If the plant stopped growing after 10 months, what would the graph look like?
5. How could you express the relationship between \( h \) and \( t \) algebraically?

**Tip:** Always check units in word problems to ensure your slope (rate of change) has the correct meaning in context.

The graph shows the height in inches, h, of a bamboo plant t months after it has been planted. What is the slope of the line? What does that value mean in context? At what point does the line intersect the h-axis? What does that value mean in this context?

Learn how to calculate the slope and interpret the intercept of a linear graph representing the growth of a bamboo plant. This problem involves key algebraic concepts, including slope and intercept, suitable for students in grades 7-8.

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