2016 TWO(e)

Excellence
Question
A cone of height hh and radius rr is inscribed, as shown, inside a sphere of radius 66 cm.
Loading diagram...
The base of the cone is ss cm below the xx-axis.

Find the value of ss which maximises the volume of the cone.
*You must use calculus and show any derivatives that you need to find when solving this problem.*

*You do not need to prove that the volume you have found is a maximum.*
Official Answer
Vol=13πr2hVol=\frac{1}{3}\pi r^2h
h=6+sh=6+s
s2+r2=62s^2+r^2=6^2
r2=36−s2r^2=36-s^2

∴V=13π(36−s2)(6+s)\therefore V=\frac{1}{3}\pi(36-s^2)(6+s)
=13π(216+36s−6s2−s3)=\frac{1}{3}\pi(216+36s-6s^2-s^3)
dVds=13π(36−12s−3s2)\frac{dV}{ds}=\frac{1}{3}\pi(36-12s-3s^2)

Max volume when dVds=0\frac{dV}{ds}=0
⇒3s2+12s−36=0\Rightarrow 3s^2+12s-36=0
s2+4s−12=0s^2+4s-12=0
(s+6)(s−2)=0(s+6)(s-2)=0
s=−6s=-6 or s=2s=2
s=2s=2
Grading Criteria

Achievement (u)

-

Merit (r)

  • Correct expression for dVds\frac{dV}{ds}

Excellence T1

  • Correct solution.

Excellence T2

-
Exam Paper (2016)Assessment Schedule (2016)
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