Exercise 81 Page 476

Compound interest is the interest earned on the principal and on previously earned interest. Let's recall the formula that gives the balance A of an account earning compound interest. A= P( 1+r/n )^(n t) In this formula, P is the principal, or initial amount, r is the annual interest rate written in decimal form, t is the time in years, and n is the number of times the interest is compounded in one year. Let's carefully consider the given exercise.

$ 1000 is deposited into a savings account that pays 6 % annual interest compounded monthly.

We can immediately identify P as 1000. The annual interest rate, written as a decimal number, is 0.06. Finally, since the interest is compounded monthly and there are 12 months in one year, we have that n= 12. We want to calculate how long we have to wait until the account is worth A = 4000. Let's substitute these values into the formula and simplify.

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

SubstituteValues

Substitute values

4000= 1000( 1+0.06/12 )^(12t)

CalcQuot

Calculate quotient

4000=1000(1+0.005)^(12t)

AddTerms

Add terms

4000=1000(1.005)^(12t)

DivEqn

.LHS /1000.=.RHS /1000.

4=1.005^(12t)

Now we want to solve the simplified equation. To do so, we will use logarithm, since the logarithm is the inverse function of exponentiation used in the equation. Let's do it!

4=1.005^(12t)

log_(1.005)(LHS)=log_(1.005)(RHS)

log_(1.005)4 = log_(1.005)1.005^(12t)

log_(1.005)(1.005^m)=m

log_(1.005)4 = 12t

Calculate logarithm

277.951443 ... = 12t

RoundInt

Round to nearest integer

278 ≈ 12t

DivEqn

.LHS /12.=.RHS /12.

278/12 ≈ 12t/12

CalcQuot

Calculate quotient

23.166666 ... ≈ t

RoundInt

Round to nearest integer

23≈ t

RearrangeEqn

Rearrange equation

t ≈ 23

We found that t is approximately 23. We have to wait about 23 years until the account is worth $ 4000.