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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