To analyze the table and model the growth of the plant, let's break it down step by step.
Step a: Model the data with an equation
From the given data:
- For \(x = 3\), \(y = 9\)
- For \(x = 5\), \(y = 15\)
- For \(x = 7\), \(y = 21\)
- For \(x = 9\), \(y = 27\)
We notice that as \(x\) increases, \(y\) also increases, and more importantly, there seems to be a consistent pattern in the growth.
If we observe the differences:
- From \(3\) to \(5\), \(y\) increases by \(6\) (from \(9\) to \(15\))
- From \(5\) to \(7\), \(y\) increases by \(6\) (from \(15\) to \(21\))
- From \(7\) to \(9\), \(y\) increases by \(6\) (from \(21\) to \(27\))
This indicates that the relationship between \(x\) and \(y\) is linear, and the slope (rate of growth) is \(3\).
We can start from the point \( (3, 9) \) and use the point-slope form of a linear equation.
Using the point \( (3, 9) \): \[ y - 9 = 3(x - 3) \]
Now we can convert this into slope-intercept form (which is \(y = mx + b\)): \[ y - 9 = 3x - 9 \] \[ y = 3x \]
Finally, let's check if the equation holds for the points we have:
- For \(x=3, y = 3(3) = 9\)
- For \(x=5, y = 3(5) = 15\)
- For \(x=7, y = 3(7) = 21\)
- For \(x=9, y = 3(9) = 27\)
The model fits perfectly.
Step b: Predict the height of the plant at 12 months
Using the model \(y = 3x\): \[ y = 3(12) = 36 \]
Conclusion
The modeled equation for the plant's height growth is: \[ y = 3x \]
And the predicted height of the plant at 12 months is: \[ 36 \text{ cm} \]