Question

If a baseball card appreciates exponentially in value by 50% over 10 years, what is its annual appreciation (growth) rate? Please use a calculator, and answer to the nearest whole percent. ​

Answer 1

we can use hmmm any value per se for the card, hmmm say let's use $2, if the baseball card was initially 2 bucks, and after 10 years it went up in value by 50%, well 50% of 2 is 1, so the new value is 3 bucks, then

`\qquad \textit{Amount for Exponential Growth} \\\\ A=P(1 + r)^t\qquad \begin{cases} A=\textit{accumulated amount}\dotfill &3\\ P=\textit{initial amount}\dotfill &2\\ r=rate\to r\%\to (r)/(100)\\ t=years\dotfill &10\\ \end{cases} \\\\\\ 3=2(1 + (r)/(100))^(10)\implies \cfrac{3}{2}=(1 + (r)/(100))^(10)\implies \cfrac{3}{2}=((100+r)/(100))^(10) \\\\\\ \sqrt[10]{\cfrac{3}{2}}=\cfrac{100+r}{100}\implies 100\sqrt[10]{\cfrac{3}{2}}=100+r \\\\\\ 100\sqrt[10]{\cfrac{3}{2}}-100=r\implies 4\approx r`