Expert: Paul Klarreich - 2/27/2007

Question
Please help solve these.

4

A farmer buys 90 sheep in 1990. In 2005 he has x sheep, where x is the last
3 digits of your student id. (Mine is 317) Assume the population is growing or
decaying exponentially.

(a) Find an exponential function f(t)=f e (kt are powers) that models this
growth, where t is the number of years after 1990.

(b) uSE  this model to predict the size of the population in 2011.




5  Carbon dating is commonly used to determine how old an object is by
measuring the amount of carbon-14 that is left in an object as the object
decays over the years.This decay prcoeeds exponentially with half-life of
approx 5800 years.How old(to the nearest year) would carbon dating predict
a piece of bone is when the amount of carbon-14 has decayed from its
original amount of 100g to final amount of 22g ?

NOTE: Radioacitve substances decays according to exponential model

A(t) = A(0)e (-at) as powers.


Please if anyone can help.......

Answer
Questioner:   pearl
Category:  Advanced Math
 
Subject:  exponential function
Question:  Please help solve these.

4. A farmer buys 90 sheep in 1990.

>> consider that to be year zero.

In 2005 he has x sheep,

>> consider that to be year 15.

where x is the last 3 digits of your student id. (Mine is 317) Assume the population

is growing or decaying exponentially.

(a) Find an exponential function f(t)=f e (kt are powers) that models this growth, where t is the number of years after 1990.

(b) uSE  this model to predict the size of the population in 2011.

5  Carbon dating is commonly used to determine how old an object is by measuring the amount of carbon-14 that is left in an object as the object decays over the years.This decay prcoeeds exponentially with half-life of approx 5800 years.How old(to the nearest year) would carbon dating predict a piece of bone is when the amount of carbon-14 has decayed from its original amount of 100g to final amount of 22g ?

NOTE: Radioactive substances decays according to exponential model

A(t) = A(0)e (-at) as powers.
..........................................................
Hi, Pearl,

Either of these is modeled by:

N(t) = N(0) e^(kt), where  k>0 for growth, and k<0 for decay.

In your problem 4:  N(15) = 317,  and  N(0) = 90

317 = 90 e^15k

e^15k = 315/90

15k = ln(315/90)

k = ln(315/90) / 15  
Use your calculator.  Mine gives  0.0835, and your model is:

N(t) = 90 e^0.0835t

Now for part b, consider 2011 to be year 21.  Compute N(21) from that.
.......................................................
For radioactive decay, a half-life of 5800 years means:

N(5800) = N(0)/2  

N(0) e^5800k = N(0)/2

e^5800k = 1/2

5800k = - ln 2

k = -0.0001195

Now for your example, use  N(0) = 100, and N(t) = 22

22 = 100 e^-0.0001195t

e^-0.0001195t = 22/100 = 0.22

-0.0001195t = ln 0.22

-0.0001195t = -1.51412

Just divide.