Answer to Question #184713 in Math for Ojaswi Mahavir Sheth

Question

Let’s suppose a gardener planted a tomato plant and she measured and kept track of the growth of the tomato plant every other day. This gardener is very curious person, and she would like to estimate how tall her plant was on the fourth day. The table of observations looked like this :

Day Height(mm)

1 0

3 4

5 7

7 9

9 12

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c} Day & Height(mm) \\ \hline 1 & 0 \\ \hdashline 3 & 4 \\ \hdashline 5 & 7 \\ \hdashline 7 & 9 \\ \hdashline 9 & 12 \end{array}$$

The interpolation formula can be written as

$$\dfrac{y-y_1 }{x-x_1}=\dfrac{y_2-y_1}{x_2-x_1}$$

$$y=y_1 +\dfrac{y_2-y_1}{x_2-x_1}(x-x_1)$$

The first set of value for day three is given as $(3, 4).$ The second set of value for day five is given as $(5, 7).$

After substituting these given values into the formula, we can calculate the estimated height of the plant on the fourth day

$$y=4 +\dfrac{7-4}{5-3}(4-3)$$

$$y=5.5 \ mm$$

The estimated height of the plant on the fourth day is 5.5 mm.