Question

interpolating data
heather is training for a long-distance run. her data points listed below represent the days of practice, $x$, and the number of miles run, $y$
$(1, 2.5), (2, 4.2), (4, 5.6), (6, 7), (8, 8.1), (10, 11)$
use the equation to interpolate the value and estimate the distance that she could have run on day 3. round to the nearest tenth of a mile.
day 3 = ______ miles

Explanation:

Step1: Choose interpolation method

We will use linear interpolation between the points closest to day 3: $(2, 4.2)$ and $(4, 5.6)$.

Step2: Define linear interpolation formula

The linear interpolation formula for a value $x$ between $x_1$ and $x_2$ is:
$$y = y_1 + \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$
Here, $x_1=2$, $y_1=4.2$, $x_2=4$, $y_2=5.6$, $x=3$.

Step3: Calculate slope

$$\frac{y_2 - y_1}{x_2 - x_1} = \frac{5.6 - 4.2}{4 - 2} = \frac{1.4}{2} = 0.7$$

Step4: Compute interpolated y-value

$$y = 4.2 + 0.7(3 - 2) = 4.2 + 0.7(1) = 4.9$$

Answer:

4.9