Question

Can somebody help me with #3 I don’t get it ?

Answer 1

From the amortization function:

`\begin{gathered} A(t)=A_o(1+(r)/(n))^(nt) \\ where \\ A_o\Rightarrow initial\text{ amount invested} \\ r\Rightarrow annual\text{ interest rate} \\ n\Rightarrow number\text{ of times compounded} \\ t\Rightarrow time \end{gathered}`

Given that $10,000 is invested at an annual rate of 5% compounded quarterly, this implies that

`\begin{gathered} A_o=10000 \\ r=5\%=0.05 \\ n=4 \end{gathered}`

Thus, by substituting these values into the above equation, we have

`\begin{gathered} A(t)=10000(1+(0.05)/(4))^(4t) \\ \Rightarrow A(t)=10000(1.0125)^(4t) \end{gathered}`

Given that the amount earned is double the initial amount invested, this implies that

`A(t)=20000`

Thus, we have

`20000=10000(1.0125)^(4t)`

To solve for t,

`\begin{gathered} divide\text{ both sides by 10000} \\ (20000)/(10000)=(10000(1.0125)^(4t))/(10000) \\ \Rightarrow2=(1.0125)^(4t) \\ take\text{ the logarithm of both sides,} \\ \ln2=\ln(1.0125)^(4t) \\ 0.6931471806=4t*0.01242252 \\ \Rightarrow4t=55.79763048 \\ divide\text{ both sides by the coefficient of t, which is 4} \\ thus, \\ (4t)/(4)=(55.79763048)/(4) \\ \Rightarrow t=13.94940762 \end{gathered}`

Hence, it takes 13.94940762 years to double the amount invested.