Question: A scientist is studying red maple tree growth in a skate park. She measured the trunk diameters of a sample of trees in the same month every other year. The tables show the data for two of the trees.
Tree 1
Year Trunk Diameter (inches)
1 18.6
3 19.2
5 19.8
7 20.4
9 21.0
11 21.6
13 22.2
Tree 2
Year Trunk Diameter (inches)
1 11.4
3 12.0
5 12.6
7 13.2
9 13.8
11 14.4
13 15.0
This is the final year in which she will collect data. When her data collection is complete, she will predict future red maple tree growth.
The scientist needs an equation to model her data so she can predict the diameter of the future complete linear equation that can explain for tree 1, where $x$ is the year and $y$ is the tree diameter, in inches.
Look on the variables and numbers you want to know and drag them into the boxes.
Equation for Tree 1
y = [ ] [ ] \cdot [ ]
[ -0.6 ] [ -0.3 ]
[ 10.0 ] [ 10.3 ] [ 20.6 ] [ x ]
1. **State the problem:** We need to find a linear equation $y = mx + b$ that models the trunk diameter $y$ of Tree 1 as a function of the year $x$.
2. **Identify variables:** $x$ is the year, $y$ is the trunk diameter in inches.
3. **Use two data points from Tree 1 to find slope $m$:**
Choose points $(1, 18.6)$ and $(13, 22.2)$.
$$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{22.2 - 18.6}{13 - 1} = \frac{3.6}{12} = 0.3$$
4. **Find the y-intercept $b$ using point-slope form:**
$$y = mx + b \Rightarrow b = y - mx$$
Using point $(1, 18.6)$:
$$b = 18.6 - 0.3 \times 1 = 18.6 - 0.3 = 18.3$$
5. **Write the linear equation:**
$$y = 0.3x + 18.3$$
6. **Interpretation:** The trunk diameter increases by $0.3$ inches every year starting from $18.3$ inches at year zero (extrapolated).
7. **Check with another point:**
At $x=3$, predicted $y = 0.3 \times 3 + 18.3 = 0.9 + 18.3 = 19.2$, which matches the data.
**Final answer:**
$$\boxed{y = 0.3x + 18.3}$$