# Thread: Rate of change. Forming an equation

1. ## Rate of change. Forming an equation

A forest is buring so that, $t$ hours after the start of the fire, the area is $A$ hectares. It is given that, at any instant, thhe rate at which the area is increasing is proportional to $A^2$.

i) Write down a differential equation which models this situation.

$\frac{dt}{dA}= A^2$ ?

ii)After 1 hour, 1000 hetares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt.

Am i right for part i)?
Not sure how to do part ii)

2. Originally Posted by George321
A forest is buring so that, $t$ hours after the start of the fire, the area is $A$ hectares. It is given that, at any instant, thhe rate at which the area is increasing is proportional to $A^2$.

i) Write down a differential equation which models this situation.

$\frac{dt}{dA}= A^2$ ?

ii)After 1 hour, 1000 hetares have been burnt; after 2 hours, 2000 hectares have been burnt. Find after how many hours 3000 hectares have been burnt.

Am i right for part i)?
Not sure how to do part ii)
Read what the question says:

It is given that, at any instant, thhe rate at which the area is increasing is proportional to $A^2$.
So $\frac{dA}{dt} \propto A^2 \Rightarrow \frac{dA}{dt} = k A^2$ where k is a constant.

3. Why is there a K and how do you do part ii)?

4. Originally Posted by George321
Why is there a K and how do you do part ii)?
k is the proportionality constant. You're expected to be familiar with the topic 'Variation' - I suggest you review it.

For part ii), the first step is to get the general solution to the DE. Show you working. Then I'll explain the second step.