Question

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60. If a bond earns 5% interest per year compounded continuously, how many years will it take for an initial investment of $100 to $1000?

Answer 1

46.05 years

Explanation:

`\boxed{\begin{minipage}{8.5 cm}\underline{Continuous Compounding Formula}\\\\$ A=Pe^(rt)$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\\phantom{ww}$\bullet$ $P =$ principal amount \\\phantom{ww}$\bullet$ $e =$ Euler's number (constant) \\\phantom{ww}$\bullet$ $r =$ annual interest rate (in decimal form) \\\phantom{ww}$\bullet$ $t =$ time (in years) \\\end{minipage}}`

Given:

• A = $1000
• P = $100
• r = 5% = 0.05

Substitute the given values into the continuous compounding formula and solve for t.

`\implies 1000=100e^(0.05t)`

`\implies 10=e^(0.05t)`

`\implies \ln 10=\ln e^(0.05t)`

`\implies \ln 10=0.05t\ln e`

`\implies \ln 10=0.05t`

`\implies t=(1)/(0.05)\ln 10`

`\implies t=20\ln 10`

`\implies t=46.0517018... \rm years`

Therefore, it will take 46.05 years for the initial investment of $100 to reach $1000.