Solve this ch 3 pre calc question for me?

[bbtne]

In a typing class, the average number (N) of words per minute typed after t weeks of lessons was found to be N=157/1+5.4e^-0.12t
find the time necessary to type
(a)50 words per minute
(b)75 words per minute
pleeeaaaaseee help!!

 

[soroban]

Hello, bbtne!

I'll do part (a) . . .

\(\displaystyle \text{In a typing class, the average number }(N)\text{ of words per minute typed}\)
. . \(\displaystyle \text{after }t\text{ weeks of lessons was found to be: }\:N\:=\:\dfrac{157}{1+5.4e^{-0.12t}}\)
\(\displaystyle \text{Find the time necessary to type}\)

\(\displaystyle \text{(a) 50 words per minute}\)

\(\displaystyle \text{(b) 75 words per minute}\)

\(\displaystyle \text{We have: }\:50 \:=\:\dfrac{157}{1 + 5.4e^{-0.12t}} \quad\Rightarrow\quad 50(1 + 5.4e^{-0.12t}) \:=\:157 \)


. . . . . . . . \(\displaystyle 50 + 270e^{-0.12t} \:=\:157 \quad\Rightarrow\quad 270e^{-0.12t} \:=\:107 \quad\Rightarrow\quad e^{-0.12t} \:=\:\frac{107}{270} \)


Take logs: .\(\displaystyle \ln\left(e^{-0.12t}\right) \:=\:\ln\left(\frac{107}{270}\right) \quad \Rightarrow\quad -0.12t\ln(e) \:=\:\ln\left(\frac{107}{270}\right) \)

. . . . . . . . . . . \(\displaystyle -0.12t \:=\:\ln\left(\frac{107}{270}\right) \quad\Rightarrow\quad t \:=\:\dfrac{\ln(\frac{107}{270})}{-0.12} \:=\:7.713276038\)


Therefore: .\(\displaystyle t \:\approx\:7.7\text{ weeks.}\)