An oil slick of area 20m^2 tripples in size everyday. Find the time taken for it to exceed 1 hactare in size by firstly representing this information with an appropriate equation.

The oil slick area A exhibits exponential growth. Assume time t is measured in days.

dAdt=kA, A0=20, A1=20*3=60

Solve this separable differential equation.

dAdt=kA

∫dAA=∫kdt⟺ln (A) = kt+C

A (t) = A0ekt

Determine the constant k using the initial and first day oil slick area values A0, A1.

A1=A0ek*1 on day one

k=ln (A1A0) = ln (6020) = ln (3)

Substitute the known constant k and A0 into the equation.

A (t) = 20eln (3) t

Verify the model A (t) matches the desired oil slick expansion. Does it triple every day?

A (0) = 20e0=20

A (1) = 20eln (3) 1=60=3*20

A (2) = 20eln (3) 2=180=3*60

A (3) = 20eln (3) 3=540=3*180

It checks out!

On what day has the oil slick reached 1 hectare? The area is measured in square meters.

1 hectare = 10,000 square meters

10000≤20eln (3) t Solve for t days.

t≥1ln (3) ln (1000020)

t≥5.657 days

Answer

The oil slick reached 1 hectare after about 5+1/2 days.