Sits whe warning snt $4 / 9$ $\newline$ Interpolating Data $\newline$ 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, $y$ . $\newline$ $(1,2.5),(2,4.2),(4,5.6),(6,7),(8,8.1),(10,11)$ $\newline$ 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. $\newline$ day $3=$ $\square$ miles

Full solution

Q. Sits whe warning snt

4 / 9

\newline

Interpolating Data

\newline

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

\newline

(1,2.5),(2,4.2),(4,5.6),(6,7),(8,8.1),(10,11)

\newline

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.

\newline

day

3=

\square

miles

Find Data Points: First, let's find two data points around day $3$ to use for interpolation. We can use $(2, 4.2)$ and $(4, 5.6)$ .
Calculate Slope: Now, calculate the slope $m$ of the line between these two points using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ . $m = \frac{5.6 - 4.2}{4 - 2} = \frac{1.4}{2} = 0.7$
Use Point-Slope Form: Next, use the point-slope form of the equation of a line, $y - y_1 = m(x - x_1)$ , with point $(2, 4.2)$ and the slope $0.7$ to find the equation. $\newline$ $y - 4.2 = 0.7(x - 2)$
Simplify Equation: Simplify the equation to get $y$ in terms of $x$ . $\newline$ $y = 0.7x - 0.7(2) + 4.2$ $\newline$ $y = 0.7x - 1.4 + 4.2$ $\newline$ $y = 0.7x + 2.8$
Plug in $x$ : Now, plug in $x = 3$ to estimate the distance for day $3$ . $\newline$ $y = 0.7(3) + 2.8$ $\newline$ $y = 2.1 + 2.8$ $\newline$ $y = 4.9$