NCEA Calculus 2013 ONE(e - Solution & Explanation

Question

A closed cylindrical tank is to have a surface area of

20\ \mathrm{m}^2

.

Find the radius the tank needs to have so that the volume it can hold is as large as possible.

*You do not have to prove that your solution gives the maximum volume.*

*Show any derivatives that you need to find when solving this problem.*

Official Answer

20 = 2\pi r^2 + 2\pi rh

2\pi r\,(r+h)=20

h=\dfrac{10}{\pi r}-r

V=\pi r^2 h=\pi r^2\cdot\left(\dfrac{10}{\pi r}-r\right)

V=10r-\pi r^3

\dfrac{dV}{dr}=10-3\pi r^2

\dfrac{dV}{dr}=0\ \Rightarrow\ r=\sqrt{\dfrac{10}{3\pi}}\ \text{or}\ r=1.03\text{ m}

OR

20=\pi r^2+2\pi rh

V=10r-\dfrac{\pi r^3}{2}

\dfrac{dV}{dr}=10-\dfrac{3\pi r^2}{2}

r=\sqrt{\dfrac{20}{3\pi}}=1.46

Grading Criteria

Achievement (u)

-

Merit (r)

Equation for volume in terms of $1$ variable found, and differentiated correctly.

Excellence T1

Problem solved including correct derivative.

Excellence T2

-

2013 ONE(e)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2