Chapter 4.3, Problem 10GP

Solution

(a.)

The balance after $2$ years.

It has been determined that the balance in the account after $2$ years is $\$\text{2762}\text{.93}$.

Given:

An amount of $\$2500$ is deposited in an account that pays $5\%$ annual interest compounded continuously.

Concept used:

If interest is compounded continuously, then the amount $A$ in an account after $t$ years is given as $A=P{e}^{rt}$ , where $P$ is the principal and $r$ is the annual interest rate expressed as a decimal.

Calculation:

It is given that an amount of $\$2500$ is deposited in an account that pays $5\%$ annual interest compounded continuously.

Then, $P=2500$ and $r=5\%$ , which simplifies to $r=0.05$.

Put the above values and $t=2$ in $A=P{e}^{rt}$ to get,

  $\begin{array}{c}A=2500e^{0.05\left(2\right)}\\ =2500e^{0.1}\\ \approx 2500\left(\text{1}\text{.10517091}\right)\\ \approx \text{2762}\text{.93}\end{array}$

This implies that the balance in the account after $2$ years is $\$\text{2762}\text{.93}$.

Conclusion:

It has been determined that the balance in the account after $2$ years is $\$\text{2762}\text{.93}$.

(b.)

The balance after $5$ years.

It has been determined that the balance in the account after $5$ years is $\$\text{3210}\text{.06}$.

Given:

An amount of $\$2500$ is deposited in an account that pays $5\%$ annual interest compounded continuously.

Concept used:

If interest is compounded continuously, then the amount $A$ in an account after $t$ years is given as $A=P{e}^{rt}$ , where $P$ is the principal and $r$ is the annual interest rate expressed as a decimal.

Calculation:

It is given that an amount of $\$2500$ is deposited in an account that pays $5\%$ annual interest compounded continuously.

Then, $P=2500$ and $r=5\%$ , which simplifies to $r=0.05$.

Put the above values and $t=5$ in $A=P{e}^{rt}$ to get,

  $\begin{array}{c}A=2500e^{0.05\left(5\right)}\\ =2500e^{0.25}\\ \approx 2500\left(\text{1}\text{.28402541}\right)\\ \approx \text{3210}\text{.06}\end{array}$

This implies that the balance in the account after $5$ years is $\$\text{3210}\text{.06}$.

Conclusion:

It has been determined that the balance in the account after $5$ years is $\$\text{3210}\text{.06}$.

(c.)

The balance after $7.5$ years.

It has been determined that the balance in the account after $7.5$ years is $\$\text{3637}\text{.48}$.

Given:

An amount of $\$2500$ is deposited in an account that pays $5\%$ annual interest compounded continuously.

Concept used:

If interest is compounded continuously, then the amount $A$ in an account after $t$ years is given as $A=P{e}^{rt}$ , where $P$ is the principal and $r$ is the annual interest rate expressed as a decimal.

Calculation:

It is given that an amount of $\$2500$ is deposited in an account that pays $5\%$ annual interest compounded continuously.

Then, $P=2500$ and $r=5\%$ , which simplifies to $r=0.05$.

Put the above values and $t=7.5$ in $A=P{e}^{rt}$ to get,

  $\begin{array}{c}A=2500e^{0.05\left(7.5\right)}\\ =2500e^{0.375}\\ \approx 2500\left(\text{1}\text{.45499141}\right)\\ \approx \text{3637}\text{.48}\end{array}$

This implies that the balance in the account after $7.5$ years is $\$\text{3637}\text{.48}$.

Conclusion:

It has been determined that the balance in the account after $7.5$ years is $\$\text{3637}\text{.48}$.