Question

The hourly wage of some automobile plan workers went from $7.50 to $12.22 in 10 years (annual raises).If their wages are growing exponentially what will be their hourly wage in 10 more years?(nearest $0.10)

Answer 1

Define the exponential function

$A(t)=A_0e^{rt}$

where A is the hourly wage.

Given that A(0) = 7.50 and A(10) = 12.22.

By using definition of A(t),

$\begin{gathered} 7.50=A_0e^0 \\ A_0=7.50 \end{gathered}$

Then

$A(t)=7.5e^{rt}$

Now, use A(10)=12.22.

$\begin{gathered} 7.5e^{10r}=12.22 \\ e^{10r}=1.629 \end{gathered}$

Take logarithm on both sides.

$\begin{gathered} 10r=\ln (1.629) \\ =0.488 \\ r=0.0488 \end{gathered}$

Substitute the value of r in A(t).

$A(t)=7.5e^{0.0488t}$

Find the wage after 10 more years.

Take t = 20 and substitute into A(t).

$\begin{gathered} A(20)=7.5e^{0.0488\cdot20} \\ =19.904 \end{gathered}$

In 20 years, the daily wage will be $19.904.