I am doing Cambridge AS level maths past papers and came across a question who's answer I don't understand.
The question is:
An oil pipeline under the sea is leaking oil and a circular patch of oil has formed on the surface of the sea. At midday the radius of the patch of oil is $50~\text{m}$ and is increasing at a rate of $3~\text{m}$ per hour. Find the rate at which the area of the oil is increasing at midday.
The answer is:
$A= \pi r^2$ leads to $\frac{dA}{dr} = 2\pi r$
it then continues by using the chain rule and the fact that $\frac{dr}{dt}= 3$ which I all understand.
But why is $\frac{dA}{dr} =2\pi r$?
Best Answer
Does the the following statement make sense to you?
If $y=x^2$ then ${dy}/{dx}=2x$
Multiply both side by $pi$:
If $y=\pi x^2$ then ${dy}/{dx}=2\pi x$
Substitute $A$ for $y$ and $r$ for $x$...
If $A=\pi r^2$ then ${dA}/{dr}=2\pi r$
or, in other words,
$A=\pi r^2$ would lead to ${dA}/{dr}=2\pi r$