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Flashcards in Chapter 3 Deck (19)
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1
Q

Radioactivity

General Differential Equation

A

dN/dt = -kN

2
Q

Radioactivity

General Equation

A

N = N0 e^-k(t-t0)

3
Q

Half Life

Definition

A

the time after which half of the isotope has decayed

4
Q

Half Life

General Equation

A

Th = 1/k log2

5
Q

What is the half life of Carbon-14?

A

5730 years

6
Q

General Population Growth Equation

A

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)
7
Q

Population Growth

The Malthusian Model - Differential Equation

A
-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
8
Q

Population Growth

The Malthusian Model - General Solution

A

p(t) = p(t0)*e^γ(t-t0)

9
Q

Population Growth

The Malthusian Model - Growth Rate

A

if γ>0, the population will increase without bound
if γ=0, population size is constant
if γ<0, population will decrease to 0

10
Q

Population Growth

The Logistic Growth - Differential Equation

A

-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)

11
Q

Newton’s Law of Cooling - Differential Equation

A

dθ/dt = -k (θ-A)
with θ(t0) = θ0

θ = object's temperature
A = ambient temperature
12
Q

Newton’s Law of Cooling - General Equation

A

θ(t) = A + (θ0 - A)e^-k(t-t0)

13
Q

Mixing Problems - Differential Equation

A

dN/dt = rinCin - routCout

N = amount of pollutant
rin = rate of inflow
Cin = concentration of inflow
rout = rate of outflow
Cout = concentration of outflow
14
Q

Mixing Problems - Concentration

A

C = n/V

15
Q

Supply and Demand - Differential Equation

A

dP/dt = E (Qd - Qs)

With P(0) = P0

Qd = A - BP
Qs = C + DP
P = price
Qs = supply
Qd = demand
16
Q

Supply and Demand - General Equation

A

P(t) = A-C/B+D + (P0 - A-C/B+D)e^-E(B+D)t

17
Q

Continuously Compounded Interest - Word Equation

A

rate of change of money in account = rate of accrual of interest in the account - rate of withdrawal of money from the account

18
Q

Continuously Compounded Interest - Differential Equation

A

dM/dt = rM - I

r = interest rate
M = money in account
I = income taken from acoutn
19
Q

Continuously Compounded Interest - General Equation

A

M(t) = I/r + (M0 - I/r)e^rt

r = rate of interest
I = rate of withdrawal
M = money in account
t = time