Question

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 = $square$ miles

Explanation:

Step1: Choose linear interpolation

We use linear interpolation between the points closest to $x=3$, which are $(2, 4.2)$ and $(4, 5.6)$. The linear interpolation formula is:
$$y = y_1 + \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$
where $(x_1,y_1)=(2,4.2)$, $(x_2,y_2)=(4,5.6)$, and $x=3$.

Step2: Calculate the slope

Compute the slope between the two points:
$$\frac{y_2 - y_1}{x_2 - x_1} = \frac{5.6 - 4.2}{4 - 2} = \frac{1.4}{2} = 0.7$$

Step3: Plug values into formula

Substitute $x=3$, $y_1=4.2$, and the slope into the formula:
$$y = 4.2 + 0.7(3 - 2)$$

Step4: Compute final value

Simplify the expression to find $y$:
$$y = 4.2 + 0.7(1) = 4.9$$

Answer:

4.9