NCEA Calculus 2015 ONE(e - Solution & Explanation

Question

Salt harvested at the Grassmere Saltworks forms a cone as it falls from a conveyor belt.
The slant of the cone forms an angle of

30^\circ

with the horizontal.
The conveyor belt delivers the salt at a rate of

2\ \mathrm{m}^3

of salt per minute.

Loading diagram...

Find the rate at which the slant height is increasing when the radius of the cone is

10\ \mathrm{m}

.
*You must use calculus and show any derivatives that you need to find when solving this
problem.*

Official Answer

Let

V

= volume (

m^3

)

S

= slant height (m)

h

= height (m)

r

= radius (m)

\cos 30 = \dfrac{r}{S}

S = \dfrac{r}{\cos 30}

\dfrac{dS}{dr} = \dfrac{1}{\cos 30}

\tan 30 = \dfrac{h}{r}

h = r\tan 30

V = \dfrac{1}{3}\pi r^2 h

\quad = \dfrac{1}{3}\pi r^3\tan 30

\dfrac{dV}{dr} = \pi r^2\tan 30

\dfrac{dS}{dt} = \dfrac{dS}{dr} \times \dfrac{dr}{dV} \times \dfrac{dV}{dt}

\quad = \dfrac{1}{\cos 30} \times \dfrac{1}{\pi r^2\tan 30} \times 2

When

r = 10

m,

\dfrac{dS}{dt} = \dfrac{1}{\cos 30} \times \dfrac{1}{\pi 10^2 \times \tan 30} \times 2

\quad = 0.01273

m/minute

Grading Criteria

-

2015 ONE(e)