Question

Durante quanto tempo devo aplicar um capital de 5000 reais a 20% a.m, para obter de juro uma importância igual ao dobro do capital?

Answer 1

Para o cálculo do montante temos a seguinte equação:

$\mathsf{M=C\,.\,(1\,+\,i)^{t}}$

Do enunciado temos:

$\begin{cases}\mathsf{C\,=\,R\ \,\,\,5\,000,00}\\ \\ \mathsf{M=2\,.\,C}\\ \\ \mathsf{i=20\%\,\,\,\text{ao mes}}\end{cases}$

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Faremos:

$\mathsf{2\,.\,5\,000=5\,000\,.\,(1\,+\,0,2)^{t}}\\ \\ \\ \mathsf{2=(1,2)^{t}}\\ \\ \\ \mathsf{log\,2=log\,(1,2)^{t}}$

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Temos as seguintes propriedades para logaritmo

$\mathsf{log_{b}\,(a)^{x}=x\,.\,log_{b}\,a}\\ \\ \\ \mathsf{log_{b}\,(a\,.\,c)=log_{b}\,a\,+\,log_{b}\,c}\\ \\ \\ \mathsf{log_{b}\,\bigg(\dfrac{a}{c}\bigg)=log_{b}\,a-log_{b}\,c}\\ \\ \\ \mathsf{log_{a}\,a=1}$

Podemos escrever 1,2 da seguinte forma:

$\mathsf{1,2=\dfrac{12}{10}=\dfrac{6}{5}=\dfrac{2\,.\,3}{(10/2)}=\dfrac{2\,.\,3\,.\,2}{10}}$
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Com isso, faremos:

$\mathsf{log\,2=log\,\bigg(\dfrac{2\,.\,3\,.\,2}{10}\bigg)^{t}}\\ \\ \\ \mathsf{log\,2=t\,.\,log\,\bigg(\dfrac{2\,.\,3\,.\,2}{10}\bigg)}\\ \\ \\ \mathsf{log\,2=t\,.\,\bigg[log\,(2\,.\,3\,.\,2)-log\,10\bigg]}\\ \\ \\ \mathsf{log\,2=t\,.\,\bigg[log\,2\,+\,log\,3\,+\,log\,2-1\bigg]}\\ \\ \\ \mathsf{log\,2=t\,.\,\bigg[2\,.\,log\,2\,+\,log\,3-1\bigg]}$

Sendo $\mathsf{log\,2\approx0,3\,\,\,\,\,\text{e}\,\,\,\,\, log\,3\approx0,48}$, temos:

$\mathsf{0,3=t\,.\,[2\,.\,0,3\,+\,0,48-1]}\\ \\ \\ \mathsf{0,3=t\,(0,08)}\\ \\ \\ \mathsf{t=\dfrac{0,3}{0,08}\,\,\,\,\,\to\,\,\,\,\,\mathsf{t\approx3,75}\,\,\,\to\,\,\,\,\boxed{\mathsf{t\approx3\,\,\text{meses e}\,\,\,\,3\,\,\text{semanas}}}}$

Assim, para que o capital inicial dobre sob uma taxa de 20% ao mês teremos que esperar cerca de 3 meses e 3 semanas.