Question

PLEASE HELP ME

If Gen Z's debt increases by 67.2% per year. and their current debt is 16,000. How long would it take for Gen Z to be 140,000 in debt? Could you also please explain how you got the answer.

Answer 1

`\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$ 140000\\ P=\textit{initial amount}\dotfill &16000\\ r=rate\to 67.2\%\to (67.2)/(100)\dotfill &0.672\\ t=years\\ \end{cases}`

`140000 = 16000(1 + 0.672)^(t) \implies \cfrac{140000}{16000}=1.0672^t\implies \cfrac{35}{4}=1.0672^t \\\\\\ \log\left( \cfrac{35}{4} \right)=\log(1.0672^t)\implies \log\left( \cfrac{35}{4} \right)=t\log(1.0672) \\\\\\ \cfrac{\log\left( (35)/(4) \right)}{\log(1.0672)}=t\implies 33.35\approx t\qquad \textit{about 33 years and 128 days}`