Explanation:
The general form of an exponential funtion isgiven below as
[tex]y=ab^x[/tex]
From the question, the two points given are
[tex]\begin{gathered} (2.5,13) \\ (9.5,20) \end{gathered}[/tex]
Step 1:
Put x=2.5 and y=13 to get equation 1
[tex]\begin{gathered} y=ab^x \\ 13=ab^{2.5}----(1) \end{gathered}[/tex]
Step 2:
Put x=9.5 and y=20 get equation 2
[tex]\begin{gathered} y=ab^x \\ 20=ab^{9.5}-----(2) \end{gathered}[/tex]
Step 3:
Divide equation (2) by (1)
[tex]\begin{gathered} \frac{20}{13}=\frac{ab^{9.5}}{ab^{2.5}} \\ b^7=\frac{20}{13} \\ b=\sqrt[7]{\frac{20}{13}} \end{gathered}[/tex]
Sustitute b in equation 1
[tex]\begin{gathered} 13=ab^{2.5} \\ a=\frac{13}{(\sqrt[7]{\frac{20}{13})^{2.5}}} \\ a=11.146 \end{gathered}[/tex]
Step 4:
Find y when x=16
[tex]\begin{gathered} y=ab^x \\ y=11.146(\sqrt[7]{\frac{20}{13})^{16}} \\ y=29.84 \end{gathered}[/tex]
Hence,
The final answer is
[tex]f(16)=29.84[/tex]
