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 = miles

Explanation:

Step1: Find the linear - interpolation formula

The linear interpolation formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ for a value $x$ is $y=y_1+\frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$. We want to find the value of $y$ when $x = 3$. The two points that $x = 3$ lies between are $(2,4.2)$ and $(4,5.6)$. Here $x_1 = 2$, $y_1=4.2$, $x_2 = 4$, $y_2 = 5.6$.

Step2: Substitute the values into the formula

Substitute $x_1 = 2$, $y_1=4.2$, $x_2 = 4$, $y_2 = 5.6$ and $x = 3$ into the formula:
\[

$$\begin{align*} y&=4.2+\frac{(3 - 2)(5.6 - 4.2)}{4 - 2}\\ &=4.2+\frac{1\times1.4}{2}\\ &=4.2 + 0.7\\ &=4.9 \end{align*}$$

\]

Answer:

$4.9$