Question

The function V (t) = 102(1.0059)^4t

gives the value, V, in dollars, of an account after t years.
Which of the following is the annual interest rate of this account to the nearest hundredth of a percent?

Answer 1

To find the annual interest rate of the account expressed by
`V(t) = 102(1.0059)^{4t`, the formula for compound interest is used to first determine the quarterly growth rate, which is then converted to the annual rate by raising it to the power of 4 and subtracting 1, resulting in approximately 2.37%.

The student's question involves finding the annual interest rate of an account using the function
`V(t) = 102(1.0059)^{4t`, which gives the value of an account after t years. To find this rate, we need to understand that the function indicates the account is compounded quarterly since the base of the exponent, 1.0059, is raised to the power of 4t. So for one year (t=1), the account is compounded.

The formula for compound interest is:

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 told that the annual compounding is quartered; hence, we have n=4. Breaking down the given function, 1.0059 is the quarterly growth rate over a year, so we have (1 + r/n) = 1.0059. To find the annual rate, we need to hyphenate the quarterly rate to a yearly rate:

(1 + r) = (1.0059)4

Now we solve for r:

1 + r = (1.0059)4

r = (1.0059)4 - 1

Calculating this gives us:

r = 1.023736 - 1

= 0.023736 or 2.37%

Thus, to the nearest hundredth of a percent, the annual interest rate for this account is 2.37%.