<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1768025188494-112sf7xy8y4d.jpg" style="max-width: min(250px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram shows the cross-section of a conical vessel in the shape of an inverted right circular cone $OAB$ of height $60$ cm and base radius $18$ cm. A cylindrical container $PQRS$ of radius $r$ cm and height $h$ cm is fitted into the cone such that the upper edge of the cylinder is in contact with the vessel.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that the volume, $V$ cm$^3$, of the cylinder is given by $V = \pi \left( 60r^2 - \frac{10}{3}r^3 \right)$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that $r$ can vary, find the radius of the largest cylinder which can fit into the vessel and hence show that the cylinder occupies $\frac{4}{9}$ of the volume of the cone.</span></div></li></ol>

The diagram shows the cross-section of a conical vessel in the shape of an inverted right circular cone $OAB$ of height $60$ cm and base radius $18$ cm. A cylindrical container $PQRS$ of radius $r$ cm and height $h$ cm is fitted into the cone such that the upper edge of the cylinder is in contact with the vessel.

Show that the volume, $V$ cm$^3$, of the cylinder is given by $V = \pi \left( 60r^2 - \frac{10}{3}r^3 \right)$.

Given that $r$ can vary, find the radius of the largest cylinder which can fit into the vessel and hence show that the cylinder occupies $\frac{4}{9}$ of the volume of the cone.

Applications of Differentiation

Question 772ad22a

Video Solution: