Samantha's EDEXCEL C1 AS question from Y!A

I need help with this maths questions from edexcel as level maths, core maths 1, Ex 7I Q13?


If you don't have the book then the question is:

The total surface area of a cylinder \(A\)cm2 with a fixed volume of \(1000\) cubic cm is given by the formula \(A = 2\pi x^2 + 2000/x\), where \(x\) cm is the radius. Show that when the rate of change of the area with respect to the radius is zero, \(x^3 = 500/ \pi\)

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Solution:


The expression "when the rate of change of the area with respect to the radius is zero" means when \( \frac{dA}{dx}=0\).

So differentiate \(A\) with respect to \(x\):

\[\frac{dA}{dx} = 4 \pi x + \frac{2000 (-1) }{ x^2}\]
Now we want to find \(x\) so that \(\frac{dA}{dx} = 0\) so you need to solve

\[4 \pi x + \frac{2000 (-1) }{ x^2} = 0\]
for \(x\). This rearranges to:

\[4 \pi x = . \frac{2000 }{ x^2}\]
or:

\[x^3=\frac{500 }{ \pi}\]