NCEA Calculus 2020 ONE(e - Solution & Explanation

Question

A cylinder of height

h

and radius

r

is inscribed, as shown to the
right, inside a sphere of radius 20 cm.

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Find the maximum possible volume of the cylinder.

*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

r^2+\left(\dfrac{h}{2}\right)^2=400

r^2=400-\dfrac{h^2}{4}

V_{cyl}=\pi r^2h

\qquad=\pi\left(400-\dfrac{h^2}{4}\right)h

\qquad=\pi\left(400h-\dfrac{h^3}{4}\right)

\dfrac{dV}{dh}=\pi\left(400-\dfrac{3h^2}{4}\right)

\dfrac{dV}{dh}=0\Rightarrow 400-\dfrac{3h^2}{4}=0

h=\sqrt{\dfrac{1600}{3}}=\dfrac{40}{\sqrt{3}}=23.1\,\text{cm}

r=16.3\,\text{cm}

V=\pi\times 16.3^2\times 23.1

\qquad=19\ 300\,\text{cm}^3

V=19\ 347\,\text{cm}^3

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Grading Criteria

Achievement (u)

Correct expression for $\dfrac{dV}{dh}$ or $\dfrac{dV}{dr}$

Merit (r)

Correct value of $r$ or $h$ with correct derivatives.
Units not required.

Excellence T1

Correct solution with correct derivatives.
Units not required.

Excellence T2

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Video Explanation

NCEA Level 3 Calculus Differentiation 2020 NZQA Exam - Worked Answers by infinityplusone(starts at 11:00)

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2020 ONE(e)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2