DIFFERENTIAL EQUATION APPLICATIONS

GROWTH AND DECAY5-G

Growth and Decay Model

If y is a differentiable function of t, such that y > 0 and for some constant k then ky

dt

dy

Rate of change of the variable is proportional to the variable itself.

1) If and 718.21

1lim

e

x

x

x

nt

n

rPA

1

Growth and Decay Model

C is the initial value (the amount present at time t = 0k is the constant of proportionality (k > 0 growth and k < 0 for decay)t is timeY is the amount present at time t.

ktCey

Interest Compounded Continuously

P is the initial value (the principal amount present at time t = 0)r is the interest rate ( expressed as a decimal)t is timeA is the amount present at time t.

rtPeA

2) What is the rate of growth of the population of a city whose population triples every 100 years?

3) If the initial population of bacteria is 1500 and the population quadrupled during the first two days. What is the population after 3 days?

4) The rate of decay of a radioactive substance is proportional to the amount present. Four years ago there were 12 grams of substance. Now there are 8 grams. How many grams will there be 8 years from now?

5) Find the principal, P, that must be invested at a rate of 7% APR compounded continuously so the $500,000 will be available in 20 years.

6) Let y represent the mass, in pounds, of a radioactive element whose half-life is 4000 years. If there are 200 pounds of the element in an inactive mine, how much will still remain in 1000 years?

7) A certain population increase at a rate proportional to the square root of the population. If the population goes from 2500 to 3600 in five years, what is the population at the end of t years?

HOME WORKGrowth and Decay Worksheet 5-G