Question

time
( months )
3
5
7
9
plant height
15
25
35
45

The table shows the height of a plant as it grows.
a. Model the data with an equation.
b. Based on your model, predict the height of the plant at 12 months.

Answer 1

To model the data, we can first analyze the relationship between the months and plant height. The given data points are:

| Months (x) | Height (y) |
|------------|------------|
| 3 | 15 |
| 5 | 25 |
| 7 | 35 |
| 9 | 45 |

### a. Modeling the Data

The data suggests a linear relationship between the time (in months) and the height of the plant. We can derive the equation of the line using the formula for a line in slope-intercept form:

\[
y = mx + b
\]

Where:
- \(y\) is the height of the plant
- \(x\) is the time in months
- \(m\) is the slope of the line
- \(b\) is the y-intercept

**1. Calculate the slope (m):**

The slope \(m\) can be calculated using any two points. Let's use points (3, 15) and (5, 25).

\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{25 - 15}{5 - 3} = \frac{10}{2} = 5
\]

**2. Calculate the y-intercept (b):**

Using the point (3, 15) in the equation \(y = mx + b\):

\[
15 = 5(3) + b
\]
\[
15 = 15 + b \implies b = 0
\]

**3. Equation of the Line:**

Now, we have both \(m\) and \(b\):

\[
y = 5x
\]

### b. Predict the height of the plant at 12 months.

Using our model \(y = 5x\):

\[
y = 5(12) = 60
\]

### Conclusion:

a. The equation that models the data is \(y = 5x\).

b. Based on this model, the predicted height of the plant at 12 months is **60 units** of height (assuming the units are consistent, like cm or inches).