Question

Mia invests $4,000 in a money market account that earns 4% annual interest, compounded continuously. Approximately how many years will it take her money to grow to the $6,500 she needs for her small business start-up? (Round your answer to one decimal place.)

Answer 1

Given the question in the image, the following are the solution steps to answer the question.

STEP 1: Write the given terms

`\begin{gathered} P=P_0=\text{ \$}4000 \\ r=(4)/(100)=0.04 \\ t=? \\ A=P(t)=\text{ \$}6500 \end{gathered}`

STEP 2: Write the formula for continous compound interest

`P\left(t\right)=P_0e^{\left\{rt\right\}}`

STEP 3: Substitute the given terms in step 1

`6500=4000e^(0.04t)`

STEP 4: Solve for t

`\begin{gathered} 4000e^{\left\{0.04t\right\}}=6500 \\ \mathrm{Divide\:both\:sides\:by\:}4000 \\ (4000e^(0.04t))/(4000)=(6500)/(4000) \\ e^(0.04t)=(13)/(8) \\ \\ Apply\text{ exponent rules:} \\ 0.04t=\ln \left((13)/(8)\right) \\ t=(\ln \left((13)/(8)\right))/(0.04) \\ t=12.13769\approx12.1\text{ }years \end{gathered}`

Hence, it will take her approximately 12.1 years