Mathematics, 21.03.2020 18:50 sasalove

Jaquin hopes to earn $700 in interest in 1.6 years time from $56,000 that he has available to invest. To decide if it's feasible to do this by investing in an account that compounds monthly, he needs to determine the annual interest rate such an account would have to offer for him to meet his goal. What would the annual rate of interest have to be? Round to two decimal places.

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The annual rate of interest is 1.5441% and this can be determined by using the formula of compound interest and the given data.

Given :

Jaquin hopes to earn $700 in interest in 1.6 years time from $56,000 that he has available to invest. To decide if it's feasible to do this by investing in an account that compounds monthly.

The formula of compound interest can be used to determine the annual rate of interest. The formula of compound interest is given by;

$\rm A = P(1+\dfrac{r}{n})^{nt}$

where A is the final amount, r is the interest rate, n is the number of times interest applied per time period, t is the number of time periods elapsed and P is the principal amount.

Now, substitute the known terms in the above formula.

$56700=56000(1+\dfrac{r}{12})^{12\times 1.6}$

Simplify the above expression.

$1.025=(1+\dfrac{r}{12})^{19.2}$

Take the log on both sides in the above expression.

$\rm log(1.025)=19.2\;log(1+\dfrac{r}{12})$

$\rm \dfrac{log(1.025)}{19.2}=\;log(1+\dfrac{r}{12})$

Further, simplify the above expression.

$\rm 0.0005585=log(1+\dfrac{r}{12})$

$\rm 1.001286 = 1+\dfrac{r}{12}$

$\rm 0.001286 = \dfrac{r}{12}$

0.015432 = r

To convert the value of 'r' in percentage multiply by 100.

r = 1.5441%

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Answer from: thebasedgodchri

Step-by-step explanation:

We would apply the formula for determining compound interest which is expressed as

A = P(1 + r/n)^nt

Where

A = total amount in the account at the end of t years

r represents the interest rate.

n represents the periodic interval at which it was compounded.

P represents the principal or initial amount deposited

From the information given,

P = $56,000

A = 56000 + 700 = $56700

n = 12 because it was compounded 12 times in a year.

t = 1.6 years

Therefore,

56700 = 56000(1 + r/12)^12 × 1.6

56700/56000 = (1 + r/12)^19.2

56700/56000 = (1 + 0.0833r)^19.2

1.0125 = (1 + 0.0833r)^19.2

Taking log of both sides, it becomes

Log 1.0125 = 19.2log(1 + 0.0833r)

0.005395/19.2 = log(1 + 0.0833r)

0.00028 = log(1 + 0.0833r)

Taking inverse log of both sides, it becomes

10^0.00028 = 10^log(1 + 0.0833r)

1.00064 = 1 + 0.0833r

0.0833r = 1.00064 - 1

0.0833r = 0.00064

r = 0.00064/0.0833

r = 0.0077

Multiplying by 100, it becomes

0.77%

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