At the end of the day, we're looking for ∂r.

So from your formula, ∂r ≈ dr/dθ x ∂θ

Isn't it the same???

But do note Miss Loi's earlier comment on choosing the easier expression to differentiate (i.e. dr/dθ or dθ/dr) and then 'inverting' it if necessary.

BTW, stop thinking of a fixed circle. Rather think of an elastic circular sector that can expand/contract but always having its area fixed.

You calculated and submitted your first design to your boss with angle θ and radius r.

Suddenly your boss tells you there's not enough space in the width to accommodate your θ, and you'll need to reduce your angle by 2%, while still using the same amount the materials.

After some intensive calculation and help from a sexy math tutor, you confidently tell your boss the only way to achieve this is to increase the radius of the sector by 1%, thereby lengthening the stage instead.

... after which your boss is so impressed that he started asking you for the number of your sexy math tutor.

Ahh there you go ... a true real life application for this question

er i was tinking delta 0 / 0 x 100 = -2% but from ur workings it's 2

but yet the final working is negative thus error or is my concept wrong?

PS: i tink i dun need to worry abt tis-.- rather i need to worry as to waking up on time missed my a-maths prelims paper 1 the first 40mins+ ..LOL

From your workings, it's miracle you got the correct answer even though Miss Loi didn't quite understand what you're doing there!

Yes to find your change in r (i.e. ∂r), from your small change formula i,e,

You'll first need to determine .

But like you said may not be easy to differentiate (esp with that square root), so in this case it may be easier to get instead and invert your final expression. So ...

, A represents the area.

and (quicker and simpler right?)

To find ∂θ, given θ decreased by 2%

⇒ ∂θ/θ x 100% = -2%
∂θ/θ = -2/100
∂θ = -2θ/100

So sub in all your expressions,

The percentage change in r = ∂r/r x 100%
= 1%

So your final answer is correcto but you'll need to cultivate the habit of scrutinizing the question properly before you jump in to solve it. There's no second-chance for you in the actual exam!

k i went through the question with celerity..not much thought to it..

a = (r²θ) / 2
r = √[(2a)/θ]
dr/dθ
=-r/(2 x θ) [θ is the angle)

delta r = -r/(2 x θ) x -(θ / 50)
= r / 100

percentage = delta r/ r x 100

= 1%

correcto?