heather is training for a long - distance run. her data points listed below represent the days of practice…
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
Answer
Explanation:
Step1: Find the two - closest data points
The two closest data points to $x = 3$ are $(2,4.2)$ and $(4,5.6)$.
Step2: Use the interpolation formula
The interpolation formula for a linear interpolation between two points $(x_1,y_1)$ and $(x_2,y_2)$ for a value $x$ is $y=y_1+\frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$. Here, $x_1 = 2$, $y_1=4.2$, $x_2 = 4$, $y_2 = 5.6$, and $x = 3$. First, calculate $\frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}$: [ \begin{align*} \frac{(3 - 2)(5.6 - 4.2)}{4 - 2}&=\frac{1\times1.4}{2}\ &=0.7 \end{align*} ] Then, calculate $y$: [ \begin{align*} y&=y_1+\frac{(x - x_1)(y_2 - y_1)}{x_2 - x_1}\ &=4.2+0.7\ &=4.9 \end{align*} ]
Answer:
$4.9$