3.5

Exponential andLogarithmic Models

n compoundings per year Continuous Compounding

nt

n

rPA ⎟

⎠

⎞⎜⎝

⎛ += 1 rtPeA=

An investment is made in a trust fund at an annualpercentage rate of 9.5%, compounded quarterly.How long will it take for the investment to doublein value?

t

P4

4

095.1 ⎟

⎠

⎞⎜⎝

⎛ +=P2 Divide by P

t4)02375.1(2 = Take the ln of bothsides.

t4)02375.1ln(2ln = Move the 4t out front.

)02375.1ln(42ln t= yearst 4.7)02375.1ln(4

2ln≈=

Do the same example using compounding continuously.

2P = Pe.095t

2 = e.095t

ln 2 = .095t

yearst 3.7095.

2ln≈=

Time to Double forContinuous Compounding

rt

2ln=

Rate needed to Double for Continuous Compounding

tr

2ln=

Carbon 14 C14 has a half-life of 5,730 years. If westart with 3 grams. How many grams are left aftera. 1,000 years b. 10,000 years Decay and Growth are modeled

after the equation:

A = Cekt

C = the initial amountk = rate of growth or decayt = time

First, we need to find our rate k.

Note: it takes 5,730 years for 1 g to become a half a g.

)730,5(12

1 ke= k730,52

1ln =

00012097.−=k)000,1)(00012097.(3 −= eA

gA 66.2≈a.

b.)000,10)(00012097.(3 −= eA

gA 89.≈