Question

as shown below, a right - trapezoid has two right angles (∠b and ∠c). ab = bc = 3 cm, cd = 7 cm. a solid figure is produced by rotating the trapezoid about (overrightarrow{cd}).

Explanation:

Step1: Analyze the solid - figure formed

The solid figure formed by rotating the right - trapezoid about \(CD\) is a combination of a cylinder and a cone. The cylinder has a radius \(r = 3\mathrm{cm}\) and height \(h_1=3\mathrm{cm}\), and the cone has a radius \(r = 3\mathrm{cm}\) and height \(h_2=(7 - 3)=4\mathrm{cm}\).

Step2: Calculate the volume of the cylinder

The volume formula of a cylinder is \(V_{cylinder}=\pi r^{2}h_1\). Substitute \(r = 3\mathrm{cm}\) and \(h_1 = 3\mathrm{cm}\) into the formula: \(V_{cylinder}=\pi\times3^{2}\times3=27\pi\mathrm{cm}^{3}\).

Step3: Calculate the volume of the cone

The volume formula of a cone is \(V_{cone}=\frac{1}{3}\pi r^{2}h_2\). Substitute \(r = 3\mathrm{cm}\) and \(h_2 = 4\mathrm{cm}\) into the formula: \(V_{cone}=\frac{1}{3}\pi\times3^{2}\times4 = 12\pi\mathrm{cm}^{3}\).

Step4: Calculate the volume of the solid figure

The volume of the solid figure \(V=V_{cylinder}+V_{cone}\). So \(V=(27\pi + 12\pi)=39\pi\mathrm{cm}^{3}\approx39\times3.14 = 122.46\mathrm{cm}^{3}\).

Answer:

\(39\pi\mathrm{cm}^{3}\) (or approximately \(122.46\mathrm{cm}^{3}\))