NCEA Calculus 2014 THREE(e - Solution & Explanation

Question

A cone has a radius of 20 cm and a height of 40 cm.

A cylinder fits inside the cone, as shown below.

Loading diagram...

What must the radius of the cylinder be to give the cylinder the maximum volume?
You do not need to prove that the volume you have found is a maximum.
*Show any derivatives that you need to find when solving this problem.*

Official Answer

h=40-2r

V=\pi r^2h

=\pi r^2(40-2r)

=40\pi r^2-2\pi r^3

\dfrac{dV}{dr}=80\pi r-6\pi r^2

\dfrac{dV}{dr}=0\Rightarrow 80\pi r-6\pi r^2=0

2\pi r(40-3r)=0

r=\dfrac{40}{3}\text{ or }0

r=\dfrac{40}{3}\text{ cm}

Grading Criteria

Achievement (u)

Correct derivative for an incorrect but relevant expression for $V$ .

Merit (r)

A correct expression for $\dfrac{dV}{dr}$

Excellence T1

A correct solution. Units not required.

Excellence T2

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2014 THREE(e)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2