Radioactivity
General Differential Equation
dN/dt = -kN
Radioactivity
General Equation
N = N0 e^-k(t-t0)
Half Life
Definition
the time after which half of the isotope has decayed
Half Life
General Equation
Th = 1/k log2
What is the half life of Carbon-14?
5730 years
General Population Growth Equation
dP/dt = B(p,t) - D(p,t) + M(p,t)
B(p,t) = input e.g. births D(p,t) = output e.g. deaths M(p,t) = net migration (assume M = 0) dp/dt = B(p,t) - D(p,t)
Population Growth
The Malthusian Model - Differential Equation
-suggests that birth and death rates are proportional to the population so: B(p,t) = bp(t) & D(p,t) = dp(t) -where b and d are constants dp/dt = (b-d)*p(t) = γp(t) γ = b-d is the growth rate
Population Growth
The Malthusian Model - General Solution
p(t) = p(t0)*e^γ(t-t0)
Population Growth
The Malthusian Model - Growth Rate
if γ>0, the population will increase without bound
if γ=0, population size is constant
if γ<0, population will decrease to 0
Population Growth
The Logistic Growth - Differential Equation
-takes the growth rate to be
γ = μ(1 - p/p∞)
so
dp/dt = μp (1 - p/p∞)
with p(t0) = p0
p∞ = the maximum population (carrying capacity)
μ = the growth rate of a very small population (p->0)
Newton’s Law of Cooling - Differential Equation
dθ/dt = -k (θ-A)
with θ(t0) = θ0
θ = object's temperature A = ambient temperature
Newton’s Law of Cooling - General Equation
θ(t) = A + (θ0 - A)e^-k(t-t0)
Mixing Problems - Differential Equation
dN/dt = rinCin - routCout
N = amount of pollutant rin = rate of inflow Cin = concentration of inflow rout = rate of outflow Cout = concentration of outflow
Mixing Problems - Concentration
C = n/V
Supply and Demand - Differential Equation
dP/dt = E (Qd - Qs)
With P(0) = P0
Qd = A - BP Qs = C + DP
P = price Qs = supply Qd = demand
Supply and Demand - General Equation
P(t) = A-C/B+D + (P0 - A-C/B+D)e^-E(B+D)t
Continuously Compounded Interest - Word Equation
rate of change of money in account = rate of accrual of interest in the account - rate of withdrawal of money from the account
Continuously Compounded Interest - Differential Equation
dM/dt = rM - I
r = interest rate M = money in account I = income taken from acoutn
Continuously Compounded Interest - General Equation
M(t) = I/r + (M0 - I/r)e^rt
r = rate of interest I = rate of withdrawal M = money in account t = time