Question

NO LINKS!! Given the exponential relationships below, find the following​

Answer 1

Answer:

Exponential Function

General form of an exponential function with base $e$:

$f(x)=Ae^{kx}$

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 $k$:

$\begin{aligned}f(x) & =Ae^{kx}\\\implies 56 & =(40)e^{k}\\e^k & =\dfrac{56}{40}\\k & =\ln (1.4)\end{aligned}$

$\textsf{Equation}: \quad f(x)=40e^{x\ln 1.4}$

To find the population in 10 years, substitute $x = 10$ into the found equation:

$\begin{aligned}\implies f(10)&=40e^{10\ln 1.4}\\ & =1157.01862\\ & =1157\end{aligned}$

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 $k$:

$\begin{aligned}f(x) & =Ae^{kx}\\\implies 18 & =(10)e^{k}\\e^k & =\dfrac{18}{10}\\k & =\ln (1.8)\end{aligned}$

$\textsf{Equation}: \quad f(x)=10e^{x\ln 1.8}$

To find the population in 8 years, substitute $x = 8$ into the found equation:

$\begin{aligned}\implies f(8)&=10e^{8\ln 1.8}\\ & =1101.996058\\ & =1102\end{aligned}$