Question

A coin sold for $271 in 1976 and was sold again in 1988 for $417. Assume that the growth in the value V of the collector's item was exponential.

a) Find the value k of the exponential growth rate. Assume V, = 271.
k=
(Round to the nearest thousandth.)

Answer 1

well, let's noticed, the original amount is $271 and 12 years later it turned to $417, so

`\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill & \$417\\ P=\textit{initial amount}\dotfill &271\\ r=rate\to r\%\to (r)/(100)\dotfill &0\\ t=years\dotfill &12\\ \end{cases}`

`417=271(1 + (r)/(100))^(12)\implies \cfrac{417}{271}=\left( \cfrac{100+r}{100} \right)^(12) \implies \sqrt[12]{\cfrac{417}{271}}=\cfrac{100+r}{100} \\\\\\ 100\sqrt[12]{\cfrac{417}{271}}=100+r\implies 100\sqrt[12]{\cfrac{417}{271}}-100=r\implies {\Large \begin{array}{llll} \stackrel{\%}{3.657\approx r} \end{array}}`