Area & Volume：練習問題

Calculate the volume of a solid sphere with a radius of \(6 \text{ cm}\). Leave your answer in terms of \(\pi\).

A right circular cone has a base radius of \(5 \text{ cm}\) and a slant height of \(13 \text{ cm}\). Find the volume of the cone in terms of \(\pi\).

A right circular cone is divided into three parts by two planes parallel to its base. The planes are at heights \(\frac{1}{3}h\) and \(\frac{2}{3}h\) measured from the vertex, where \(h\) is the total height of the cone. Find the ratio of the volume of the top cone to the volume of the middle frustum to the volume of the bottom frustum.

In the figure, a circle has a radius of \(10\text{ cm}\). Find the length of an arc that subtends an angle of \(72^\circ\) at the center. (Leave your answer in terms of \(\pi\))

In the figure, a solid right circular cylinder has a base radius of \(7\text{ cm}\) and a height of \(10\text{ cm}\). A hole in the shape of a smaller cylinder with base radius \(3\text{ cm}\) and height \(10\text{ cm}\) is drilled vertically through the centers of the two circular bases. Find the total surface area of the resulting solid in terms of \(\pi\).

Find the volume of a sphere with a radius of \(6 \text{ cm}\). Leave your answer in terms of \(\pi\).

The surface areas of two similar solid spheres are in the ratio \(9 : 16\). If the volume of the smaller sphere is \(108\pi \text{ cm}^3\), find the volume of the larger sphere in terms of \(\pi\).

The radius of a sphere is measured as \(5.0 \text{ cm}\) with a maximum absolute error of \(0.05 \text{ cm}\). Calculate the maximum relative error in the calculated surface area of the sphere.

A solid hemisphere has a radius of \(10 \text{ cm}\).

(a) Calculate the curved surface area of the hemisphere. Leave your answer in terms of \(\pi\).

(b) Calculate the total surface area of the hemisphere. Leave your answer in terms of \(\pi\).

A solid right prism has a base in the shape of a right-angled triangle with side lengths \( 5 \text{ cm} \), \( 12 \text{ cm} \), and \( 13 \text{ cm} \). The height of the prism is \( 10 \text{ cm} \).

(a) Find the area of the triangular base of the prism.

(b) Find the volume of the prism.

(c) Calculate the total surface area of the prism.

(d) If the prism is melted and recast into a solid cube, find the side length of the cube. (Give your answer correct to 2 decimal places.)