Question

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.​

Answer 1

Answer:

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.

Step-by-step explanation: