'n Regte sirkelvormige keël, met radius $p$ en hoogte $t$, is gemeasureer (uitsnits) uit 'n soliede sfeer (met middelpunt $C$) met 'n radius van 30 cm, soos getoon in die skets - NSC Mathematics - Question 9 - 2018 - Paper 1

From the previous result we have:

$p^{2} = 60t - t^{2}$

This expression is already in its simplest form.

To find the volume of the cone:

$V(t) = \frac{1}{3} \\pi r^{2}h$

Here substitute $r = p$ and $h = t$, giving:

$V(t) = \frac{1}{3} \\pi (p^{2})(t)$

Replacing $p^{2}$ with $60t - t^{2}$ leads us to:

$V(t) = \frac{1}{3} \\pi (60t - t^{2})t$ $V(t) = 20\\\pi t^{2} \left( t - \frac{t^{2}}{60} \right)$

To find maximum volume, we set the derivative of $V(t)$ equal to zero:

$V'(t) = 20\pi t^{2} \left( 1 - \frac{t}{30} \right)$

Setting this equal to zero:

$20\pi t^{2} (1 - \frac{t}{30}) = 0$

This gives us: $t = 0 \, \text{or} \, t = 30 \, \text{cm}$