Question

2200 dollars is placed in an account with an annual interest rate of 7.25%. how much will be in the account after 29 years, to the nearest cent?

Explanation:

Step1: Identify the formula for compound interest

The formula for compound interest (assuming annual compounding, since it's an annual interest rate and no compounding period is specified otherwise) is $A = P(1 + r)^t$, where:

• $A$ is the amount of money accumulated after $n$ years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (decimal).
• $t$ is the time the money is invested for in years.

Step2: Convert the percentage to a decimal

The annual interest rate $r = 7.25\%$. To convert this to a decimal, we divide by 100: $r=\frac{7.25}{100}=0.0725$.

Step3: Identify the values of $P$, $r$, and $t$

We know that:

• The principal amount $P = 2200$ dollars.
• The annual interest rate $r = 0.0725$.
• The time $t = 29$ years.

Step4: Substitute the values into the compound - interest formula

Substitute $P = 2200$, $r = 0.0725$, and $t = 29$ into the formula $A = P(1 + r)^t$:
$A=2200\times(1 + 0.0725)^{29}$
First, calculate $(1 + 0.0725)=1.0725$.
Then, calculate $1.0725^{29}$. Using a calculator, $1.0725^{29}\approx7.3442$.
Then, multiply by the principal: $A = 2200\times7.3442$.
$2200\times7.3442 = 2200\times7+2200\times0.3442=15400 + 757.24 = 16157.24$ (approximate value, more accurately, using a calculator for $2200\times1.0725^{29}$: $2200\times1.0725^{29}\approx2200\times7.34417\approx16157.17$)

Answer:

The amount in the account after 29 years is approximately $\$16157.17$ (or $\$16157.24$ depending on the calculator precision, but to the nearest cent, we can calculate it more accurately as follows:

Using a calculator to compute $1.0725^{29}$:

$1.0725^{29}\approx e^{29\ln(1.0725)}$

$\ln(1.0725)\approx0.0700$ (more accurately, $\ln(1.0725)\approx0.06997$)

$29\times0.06997\approx2.0291$

$e^{2.0291}\approx7.343$

Then $2200\times7.343 = 2200\times7+2200\times0.343=15400+754.6 = 16154.6$? Wait, no, I think I made a mistake in the first approximation. Let's use a calculator directly:

$1.0725^{29}$:

We can calculate step - by - step:

$1.0725^1 = 1.0725$

$1.0725^2=1.0725\times1.0725 = 1.14925625$

$1.0725^3=1.14925625\times1.0725\approx1.2320$

...

But a better way is to use a calculator. Using a financial calculator or a scientific calculator:

$1.0725^{29}\approx7.34417$

$2200\times7.34417 = 2200\times7.34417 = 16157.174$

So to the nearest cent, it's $\$16157.17$)