Chapter 12.7, Problem 41E

a.

To determine

Approximate the Cleavon account balance on his $18th$ birthday.

Answer to Problem 41E

  $\$9572.29$

Explanation of Solution

Given information:

Cleavon gradmother decided to save amount for his college education when he turned $5years$ old. She deposited $\$5000$ for $5\%$ interest compounded continuously. By the time Cleavon became $18$ he will need $\$40,000$ for college tuition.

Calculation:

Consider the interest compounded continuously formula,

  $\begin{array}{l}A=P{e}^{rt}\\ A=5000e^{0.05\times \left(18-5\right)}\\ A=5000e^{0.65}\\ A=5000\left(1+0.65+\frac{0.65}{2!}+\frac{0.65}{3!}+\frac{0.65}{4!}\right)\\ A\approx \$9572.29\end{array}$

b.

To determine

Verify the Cleavon account balance is sufficient for college tuition.

Answer to Problem 41E

NO

Explanation of Solution

Given information:

Cleavon gradmother decided to save amount for his college education when he turned $5years$ old. She deposited $\$5000$ for $5\%$ interest compounded continuously. By the time Cleavon became $18$ he will need $\$40,000$ for college tuition.

Calculation:

His account balance is $\$9572.29$ , he need $\$40,000$ for college tuition it is lack by amount more than $\$30,000$

Hence, the account balance is not sufficient.

c.

To determine

Calculate account balance is reached $\$40,000$ at the Cleavon age.

Answer to Problem 41E

  $42years$

Explanation of Solution

Given information:

Cleavon gradmother decided to save amount for his college education when he turned $5years$ old. She deposited $\$5000$ for $5\%$ interest compounded continuously. By the time Cleavon became $18$ he will need $\$40,000$ for college tuition.

Calculation:

  $\begin{array}{l}A=P{e}^{rt}\\ 40000=5000e^{0.05\times t}\\ \frac{40000}{5000}=e^{0.05\times t}\\ 8=e^{0.05\times t}\\ \ln 8=\ln e^{0.05\times t}\\ \ln 8=0.05\times t\\ t=\frac{\ln 8}{0.05}\\ t\approx 42\end{array}$

Hence, account balance is reached $\$40,000$ at the Cleavon age of $42years$ .

d.

To determine

Find the amount grandmother should invest to become $\$40,000$ at his age of $18$ year.

Answer to Problem 41E

  $\$20,882$

Explanation of Solution

Given information:

Cleavon gradmother decided to save amount for his college education when he turned $5years$ old. She deposited $\$5000$ for $5\%$ interest compounded continuously. By the time Cleavon became $18$ he will need $\$40,000$ for college tuition.

Calculation:

  $\begin{array}{l}A=P{e}^{rt}\\ 40000=P{e}^{0.05\times 13}\\ 40000=P{e}^{0.65}\\ P=\frac{40000}{e^{0.65}}\\ P\approx 20,882\end{array}$

Hence, the amount grandmother should invest to become $\$40,000$ at his age of $18$ year is

  $\$20,882$ .