Question

use the exponential equation that best fits the data (2,7),(3,10),(5,50) and (8,415) to estimate the value of y when c=6

Answer 1

To determine the exponential equation that best fits the data below:

`\begin{gathered} y=ab^x \\ (2,7)\text{ (3,10) (5,50) (8, 415)} \end{gathered}`

Inorder to convert the exponential to linear, use logarithm

`\begin{gathered} y=ab^x \\ \log y=\text{loga}+\log b^x \\ \log y=\log a+x\log b \\ \text{Let log a = c, log b = m} \\ \text{compare it to straight line equation} \end{gathered}`
`\begin{gathered} y=mx+c \\ m=\log b \\ b=10^m \\ c=\log a \\ a=10^c \end{gathered}`
`\begin{gathered} m=(n\Sigma xy-\Sigma x\Sigma y)/(n\Sigma x^2-(\Sigma x)^2) \\ \text{where n=4 , }\Sigma x=18,\text{ }\Sigma x^2=102,\text{ }\Sigma y=6.17,\text{ }\Sigma xy=34.16 \\ m=(4(34.16)-18(6.17))/(4(102)-(18)^2) \\ m=0.305 \end{gathered}`
`\begin{gathered} c=(\Sigma y-m\Sigma x)/(n) \\ c=(6.17-0.305(18))/(4) \\ c=0.172 \end{gathered}`
`\begin{gathered} b=10^m \\ b=10^(0.305) \\ b=2.02 \\ a=10^c \\ a=10^(0.172) \\ a=1.49 \end{gathered}`
`\begin{gathered} y=ab^x \\ y=1.49(2.02)^x \end{gathered}`

To estimate the value of y using the above exponential equation when x= 6

`\begin{gathered} y=1.49(2.02)^6 \\ y=1.49(67.61) \\ y=100.74 \end{gathered}`

Hence the estimate the value of y ≈ 100

Therefore the correct answer is Option D