Calculus help?

Question

1. A circular city has radius r km and average population density p people per sqaure km. In 2002, the population was 3 million, the radius was 25 km and growing at 0.1km per year. If the density was increasing at 200 people per square km per year, find the rate at which the total population of the city was growing at that time.

So I kinda set up the problem first.
population =3
r=25km
dr=0.1km/yr
dp=200people/km^2/yr

so is the population a function of r and p? I got stuck, and don't know which method i should use. Can anyone gimme some ideas?

Answer 1

I'd start by noting that the population density as a function of time (D(t)) is given by the total population (P(t)) divided by the area (A(t)). Both the area and population are also functions of time:

D(t) = P(t)/A(t)

rearranging:

D(t) * A(t) = P(t)

Now, take the derivative of both sides with respect to t:

dD(t)/dt * A(t) + dA(t)/dt * D(t) = dP(t)/dt [EQUATION 1]


We also know that:

A(t) = pi* (r(t))^2

where r(t) is the radius as a function of time.

Taking the derivative of this with respect to t:

dA(t)/dt = 2*pi*r(t)*dr(t)/dt



We are told that in year t=2002:

r(t) = 25 km

dr(t)/dt = 0.1 km/yr

Therefore in 2002, A = pi*625 km^2 and dA/dt = 5 km^2/yr

We are also told that P = 3,000,000, so the population density in 2002 was D = P/A = 3,000,000/(pi*625) = pi*4800 people/km^2.

Finally, we are also told that dD(t)/dt in 2002 was 200 people/year. We now know the value of every factor/term on the left hand side of EQUATION 1 above, so we can find the value of dP(t)/dt, which is what the question is asking for.

Plugging in the various values yields:

200 people/km^2/yr * pi*625 km^2 + 5 km^2/yr * pi*4800 people/km^2 = dP(t)/dt

125,000*pi people/year + 24,000*pi people/year = pi*149,000 people/year = dP(t)/dt

Answer 2

Clearly the population is in the form

P(r(t),p(t)) = pi * r(t) ^ 2 * p(t)

Which is the area times the average density.

You need to find dP(r(t),p(t))/dt |t=0. (*)

Steps.

1. Solve for p(0) given P(r(0),p(0))=P(25,p(0))=3 000 000

2. Then use the chain rule on P(r(t),p(t)) to find the derivative (*)

3. Substitute t = 0 with the values you have.

Answer 3

since the population density does not change, the rate of population growth is directly proportional to city size growth. So it's just ((R^2/r^2) - 1)*population, which would be some number. R is the new population, r is old population.

Answer 4

Area = pi*r^2
NUmber of people = n
Pop. Density = p = # people / area = n / (pi*r^2)

At t=0 yr, n = 3000000 people and r = 25 km

dr/dt = +0.1 km/yr
dp/dt = +200 people /km^2 /yr
dn/dt = ??

p = n / (pi*r^2)

Use quotient rule

dp/dt = [(pi*r^2)*(dn/dt) - (n)(2*pi*r*(dr/dt))] / [(pi*r^2)^2]

dp/dt = +200 =
[(pi*(25 )^2)*(dn/dt) - (3000000)(2*pi*(25*(0.1))] / [(pi*25^2)^2]

dn/dt = 416,699 people / yr

Answer 5

T = Total population

T = 3.1415 * p * r^2

dT/dt = 3.1415 * (2p*dr/dt + r^2 *dp/dt)

3,000,000 = 3.1415 * p * 25^2

Solve for p and then plug into the dT/dt equation