Answer:
Exponential Function
General form of an exponential function with base [tex]e[/tex]:
[tex]f(x)=Ae^{kx}[/tex]
where:
- A is the y-intercept
- e (Euler's number) is the base
- k is some constant
Question 30
The curve crosses the y-axis at y = 40. Therefore, A = 40.
Substitute the found value of A into the formula along with (1, 56) and solve for [tex]k[/tex]:
[tex]\begin{aligned}f(x) & =Ae^{kx}\\\implies 56 & =(40)e^{k}\\e^k & =\dfrac{56}{40}\\k & =\ln (1.4)\end{aligned}[/tex]
[tex]\textsf{Equation}: \quad f(x)=40e^{x\ln 1.4}[/tex]
To find the population in 10 years, substitute [tex]x = 10[/tex] into the found equation:
[tex]\begin{aligned}\implies f(10)&=40e^{10\ln 1.4}\\ & =1157.01862\\ & =1157\end{aligned}[/tex]
Question 31
The curve crosses the y-axis at y = 10. Therefore, A = 10.
Substitute the found value of A into the formula along with (1, 18) and solve for [tex]k[/tex]:
[tex]\begin{aligned}f(x) & =Ae^{kx}\\\implies 18 & =(10)e^{k}\\e^k & =\dfrac{18}{10}\\k & =\ln (1.8)\end{aligned}[/tex]
[tex]\textsf{Equation}: \quad f(x)=10e^{x\ln 1.8}[/tex]
To find the population in 8 years, substitute [tex]x = 8[/tex] into the found equation:
[tex]\begin{aligned}\implies f(8)&=10e^{8\ln 1.8}\\ & =1101.996058\\ & =1102\end{aligned}[/tex]
