Question

Angus has $3,000 he want to invest. What interest rate compounded continuously does an account need to offer so that Angus has $5,500 in 8 years?

Answer 1

The interest rate is 7.58%

Explanation:

Compound continuous interest can be calculated using the formula:

A = P
`e^(rt)`, where

• A is the future value of the investment, including interest

• P is the principal investment amount (the initial amount)

• r is the interest rate in decimal

• t is the time the money is invested for

∵ Angus has $3,000 he want to invest

∴ P = 3000

∵ The interest rate is compounded continuously

∵ Angus has $5,500 in 8 years

∴ A = 5500

∴ t = 8

→ Substitute them in the rule above to find r

∵ 5500 = 3000
`e^(8r)`

→ Divide both sides by 3000

∴
`(11)/(6)` =
`e^(8r)`

→ Insert ㏑ in both sides

∵ ㏑(
`(11)/(6)` ) = ㏑(
`e^(8r)`)

→ Remember ㏑(
`e^(n)`) = n

∴ ㏑(
`(11)/(6)` ) = 8r

→ Divide both sides by 8

∴ 0.07576697545 = r

→ Multiply it by 100% to change it to a percentage

∴ r = 0.07576697545 × 100%

∴ r = 7.576697545 %

→ Round it to the nearest hundredth

∴ r ≅ 7.58

∴ The interest rate is 7.58%