Interpolasi Linear

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materi pendukung untuk desil klik di sini dan

materi persentil klik di sini.

Interpolasi linear adalah sebuah metode yang digunakan untuk penentuan titik di antara dua buah titik yang sudah diketahui dan segaris.

Perhatikanlah ilustrasi gambar berikut

dengan proses seperti menentukan persamaan garis lurus diperoleh rumus:

$\begin{aligned}\displaystyle \frac{y-y_{0}}{y_{1}-y_{0}}&=\frac{x-x_{0}}{x_{1}-x_{0}}\\ y-y_{0}&=\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right )\\ y&=y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right ) \end{aligned}$.

$\LARGE\colorbox{yellow}{CONTOH SOAL}$.

$\begin{array}{ll} 1.&\textrm{Tentukan taksiran nilai dari}\\ &\textrm{a}.\quad \sqrt{5}\qquad\qquad \textrm{c}.\quad \sqrt{12}\\ &\textrm{b}.\quad \sqrt{7}\qquad\qquad \textrm{d}.\quad \sqrt{22}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{a}.\quad \textrm{Di}&\textrm{ketahui bahwa}\\ &\begin{cases} \sqrt{4} & =2 \\ \sqrt{5} & =\: \: ? \\ \sqrt{9} & =3 \end{cases}\\ y&=y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right )\\ &\approx 2+\displaystyle \frac{5-4}{9-4}(3-2)\\ &\approx2+\displaystyle \frac{1}{5}\\ &\approx2+0,2\\ &\approx\color{red}2,2 \end{aligned}\\ &\begin{aligned}\textrm{b}.\quad \textrm{Di}&\textrm{ketahui bahwa}\\ &\begin{cases} \sqrt{4} & =2 \\ \sqrt{7} & =\: \: ? \\ \sqrt{9} & =3 \end{cases}\\ y&=y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right )\\ &\approx2+\displaystyle \frac{7-4}{9-4}(3-2)\\ &\approx2+\displaystyle \frac{3}{5}\\ &\approx2+0,6\\ &\approx\color{red}2,6 \end{aligned}\\ &\begin{aligned}\textrm{c}.\quad \textrm{Di}&\textrm{ketahui bahwa}\\ &\begin{cases} \sqrt{9} & =3 \\ \sqrt{12} & =\: \: ? \\ \sqrt{16} & =4 \end{cases}\\ y&=y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right )\\ &\approx3+\displaystyle \frac{12-9}{16-9}(4-3)\\ &\approx3+\displaystyle \frac{3}{7}\\ &\approx3+0,43\\ &\approx\color{red}3,43 \end{aligned} \\ &\begin{aligned}\textrm{d}.\quad \textrm{Di}&\textrm{ketahui bahwa}\\ &\begin{cases} \sqrt{16} & =4 \\ \sqrt{22} & =\: \: ? \\ \sqrt{25} & =5 \end{cases}\\ y&=y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right )\\ &\approx4+\displaystyle \frac{22-16}{25-16}(5-4)\\ &\approx4+\displaystyle \frac{6}{9}\\ &\approx4+0,67\\ &\approx\color{red}4,67 \end{aligned} \end{array}$.

$\begin{array}{ll} 2.&\textrm{Diberikan data berikut berkaitan dengan}\\ &\textrm{penduduk di suatu daerah A}\\ &\begin{array}{|c|l|l|}\hline \textrm{Tahun}&2015&2020\\\hline \begin{aligned}&\textrm{Jumlah jiwa}\\ &\textrm{daerah A} \end{aligned}&340.000&600.000\\\hline \end{array}\\ &\textrm{Tentukan perkiraan jumlah penduduk}\\ &\textrm{daerah A saat tahun 2018}\\\\ &\color{blue}\textrm{Jawab}:\\ &\begin{aligned}\textrm{Di}&\textrm{ketahui bahwa}\\ &\begin{cases} \sqrt{2015} & =340.000 \\ \sqrt{2018} & =\qquad ? \\ \sqrt{2020} & =600.000 \end{cases}\\ y&=y_{0}+\frac{x-x_{0}}{x_{1}-x_{0}}\left ( y_{1}-y_{0} \right )\\ &\approx 340.000+\displaystyle \frac{(2018-2015)}{(2020-2015)}(600.000-340.000)\\ &\approx 340.000+\displaystyle \frac{3}{5}\left ( 260.000 \right )\\ &\approx 340.000+156.000\\ &\approx\color{red}496.000 \end{aligned} \end{array}$.