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Problem 869

Find the volume of a sphere of radius \(2 .\)

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The volume of the sphere with radius 2 is V = \(\frac{32}{3}\pi\) cubic units.

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Chapter 50

A manufacturer wishes to change the container in which his product is marketed. The new container is to have the same volume as the old, which is a right circular cylinder with altitude 6 and base radius 3 . The new container is to be composed of a frustum of a right circular cone surmounted by a frustum of another right circular cone so that the smaller bases of the two frustums coincide. The radii of the bases of the lower frustum are \(3(1 / 2)\) and 2 , respectively, and its altitude is 4 . The radius of the upper base of the upper frustum is 3 . Find, to the nearest tenth, the altitude of the upper frustum.

Chapter 50

A cone is generated by rotating a right triangle with sides 3 , 4 , and 5 about the leg whose measure is 4 . Find the total area and volume of the cone.

Chapter 50

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.

Chapter 50

A metal sphere is melted and recast into a hollow spherical shell whose outer radius is \(277 \mathrm{~cm}\). The radius of the hollow interior of the shell is equal to the radius of the original sphere. Find, to the nearest centimeter, the radius of the original sphere.

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 $\mathrm{V}=(1 / 3)\left(\pi \mathrm{r}^{3}\right)(2+\sqrt{3})$.

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