Question

A person invests 8000 dollars in a bank. the bank pays 5.25% interest compounded annually. to the nearest tenth of a year, how long must the person leave the money in the bank until it reaches 12700 dollars? a, equals, p, left bracket, 1, plus, start fraction, r, divided by, n, end fraction, right bracket, start superscript, n, t, end superscript a=p(1 n r ​ ) nt

Answer 1

To find the time required for an $8,000 investment to grow to $12,700 at an annual interest rate of 5.25%, the compound interest formula is rearranged to solve for time, t. The formula used is A = P(1 + r/n)^(nt), and the calculated expression to find t is t = ln(12700/8000) / (ln(1 + 0.0525)).

Step-by-step explanation:

To calculate how long it will take for an investment of $8,000 to grow to $12,700 with an interest rate of 5.25%, compounded annually, we can use the compound interest formula A = P(1 + r/n)^(nt), 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).
• n is the number of times that interest is compounded per year.
• t is the time the money is invested for, in years.

In this case, we are solving for t when A is $12,700, P is $8,000, r is 0.0525 (5.25%), and n is 1 since the interest is compounded annually. We rearrange the formula to solve for t and get:

t = ln(A/P) / (n * ln(1 + r/n))

We then plug in the values and calculate:

t = ln(12700/8000) / (1 * ln(1 + 0.0525))

By calculating the above expression we can find the number of years required to reach the goal amount.

Answer 2

To find the time t for an investment of $8,000 to grow to $12,700 at an annual compound interest rate of 5.25%, we use the compound interest formula and solve for t, which is approximately 10.2 years when rounded to the nearest tenth.

Step-by-step explanation:

To calculate how long the $8,000 investment will take to grow to $12,700 with an annual compound interest rate of 5.25%, we can use the compound interest formula:

A = P(1 + \frac{r}{n})^{nt}

Where:

Here, A is $12,700, P is $8,000, r is 0.0525 (5.25% expressed as a decimal), and n is 1 because the interest is compounded annually. We're solving for t.

Substituting the known values into the formula, we get:

$12,700 = $8,000(1 + 0.0525)^t

To solve for t, we must isolate it on one side of the equation. We'll start by dividing both sides by $8,000:

1.5875 = (1 + 0.0525)^t

Next, we take the natural logarithm of both sides to remove the exponent on the right:

ln(1.5875) = ln((1 + 0.0525)^t)

ln(1.5875) = t * ln(1 + 0.0525)

By dividing both sides by ln(1 + 0.0525), we can solve for t:

t = \frac{ln(1.5875)}{ln(1 + 0.0525)}

Using a calculator, we find that t is approximately 10.24 years.

Therefore, it will take about 10.2 years for the $8,000 investment to grow to $12,700 at an annual compound interest rate of 5.25%, when rounded to the nearest tenth of a year.