interpolating data\nheather is training for a long - distance run. her data points listed below represent…

interpolating data\nheather 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.\n(1, 2.5), (2, 4.2), (4, 5.6), (6, 7), (8, 8.1), (10, 11)\nuse 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.\nday 3 ≈ miles

interpolating data\nheather 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.\n(1, 2.5), (2, 4.2), (4, 5.6), (6, 7), (8, 8.1), (10, 11)\nuse 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.\nday 3 ≈ miles

Answer

Explanation:

Step1: Assume a linear - interpolation

We have two data - points closest to (x = 3), which are ((2,4.2)) and ((4,5.6)). 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{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), and (x = 3).

Step2: Calculate the slope

First, calculate the slope (m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{5.6 - 4.2}{4 - 2}=\frac{1.4}{2}=0.7).

Step3: Calculate the interpolated value

Then, use the formula (y=y_1+m(x - x_1)). Substitute (x_1 = 2), (y_1 = 4.2), (m = 0.7), and (x = 3) into the formula: (y=4.2+0.7\times(3 - 2)=4.2 + 0.7=4.9).

Answer:

4.9