Question

a person invested $330 in an account growing at a rate allowing the money to double every 8 years. Write a function showing the amount of money in the account after tt years, where the annual growth rate can be found from a constant in the function. Round all coefficients in the function to four decimal places. Also, determine the percentage of growth per year, to the nearest hundredth of a percent.

Answer 1

To find the amount of money in an account after t years with a certain growth rate, use the formula A = P(1 + r/n)^(nt), where A is the amount of money, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years. In this case, the principal amount is $330 and the interest rate is 100%. The growth rate per year is 12.5%.

Step-by-step explanation:

To find the amount of money in an account after t years with an annual growth rate that allows the money to double every 8 years, we can use the formula A = P(1 + r/n)^(nt), where A is the amount of money, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

In this case, the principal amount is $330, the interest rate is 100% (since the money doubles every 8 years), and it is compounded once a year. So the formula becomes A = 330(1 + 1/8)^(8t).

To determine the percentage of growth per year, we can use the formula (r/n) * 100, where r is the annual interest rate and n is the number of times the interest is compounded per year. In this case, the growth rate per year is 100/8 = 12.5%.

Answer 2

The percentage of growth is of 9.05%.

The amount of money in the account after t years is given by
`P(t) = 330(1.0905)^t`

Step-by-step explanation:

Amount of money after t years:

The amount of money in an account after t years is given by:

`P(t) = P(0)(1+r)^t`

In which P(0) is the initial deposit and r is the growth rate, as a decimal.

Money doubles every 8 years.

This means that
`P(8) = 2P(0)`. So

`P(t) = P(0)(1+r)^t`

`2P(0) = P(0)(1+r)^8`

`(1+r)^8 = 2`

`\sqrt[8]{(1+r)^8} = \sqrt[8]{2}`

`1 + r = 2^{(1)/(8)}`

`1 + r = 1.0905`

`r = 1.0905 - 1`

`r = 0.0905`

The percentage of growth is of 0.0905 = 9.05%.

Person invested $330

This means that
`P(0) = 330`

So

`P(t) = P(0)(1+r)^t`

`P(t) = 330(1+0.0905)^t`

`P(t) = 330(1.0905)^t`

The amount of money in the account after t years is given by
`P(t) = 330(1.0905)^t`