Chapter 50: Chapter 50

In the accompanying figure, a right circular cone is constructed on the base of a hemisphere. The surface of the hemisphere is equal to the lateral surface of the cone. Show that the volume \(\mathrm{V}\) of the solid formed can be found by the formula $\mathrm{V}=(1 / 3)\left(\pi \mathrm{r}^{3}\right)(2+\sqrt{3})$.

An isosceles triangle, each of whose base angles is \(\theta\) and whose legs are \(\mathrm{a}\), is rotated through \(180^{\circ}\), using as an axis its altitude to the base. (a) Find the volume \(\mathrm{V}\) of the resulting solid, in terms of \(\theta\) and a. (b) Using the formula found in part a, find \(\mathrm{V}\), to the nearest integer if \(\mathrm{a}=5.2\) and \(\theta=27^{\circ}\). (Use \(\pi=3.14\) )

Each leg of an isosceles triangle is \(3 \mathrm{~m}\) units in length and the base is \(2 \mathrm{~m}\) units. A line \(\mathrm{s}\) is drawn through the vertex of the triangle parallel to the base. The triangle is revolved through \(360^{\circ}\) about line s with the base of the triangle always remaining parallel to s. Find, in terms of \(\mathrm{m}\), the volume of the resulting solid.

In the accompanying figure, \(A B C D\) is a trapezoid. $\mathrm{m} \angle \mathrm{A}=\mathrm{m} \angle \mathrm{B}=90^{\circ}, \mathrm{AD}=\mathrm{D} \mathrm{C}=\mathrm{m}\(, and \)\mathrm{AB}=2 \mathrm{~m}$. The line \(\mathrm{t}\), which is in the plane of \(\mathrm{ABCD}\), is parallel to \(\underline{\mathrm{AD}}\) and is \(\mathrm{m}\) units from \(\mathrm{AD}\). If the region \(\mathrm{ABCD}\) is revolved through \(360^{\circ}\) about line \(\mathrm{t}\), a solid is generated. Express in terms of \(\mathrm{m}\), the volume of this solid.