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}). (1) what is the volume of this solid figure? a (36pi cm^{3}) b (39pi cm^{3})

Explanation:

Step1: Analyze the solid - figure

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

Step2: Calculate the volume of the cylinder

The volume formula of a cylinder is $V_1=\pi r^{2}h_1$. Substitute $r = 3$ cm and $h_1 = 3$ cm into the formula: $V_1=\pi\times3^{2}\times3=27\pi$ $cm^{3}$.

Step3: Calculate the volume of the cone

The volume formula of a cone is $V_2=\frac{1}{3}\pi r^{2}h_2$. Here, $r = 3$ cm and $h_2=7 - 3 = 4$ cm. So $V_2=\frac{1}{3}\pi\times3^{2}\times4 = 12\pi$ $cm^{3}$.

Step4: Calculate the total volume of the solid figure

The volume of the solid figure $V=V_1 + V_2$. Substitute $V_1 = 27\pi$ $cm^{3}$ and $V_2 = 12\pi$ $cm^{3}$ into the formula: $V=27\pi+12\pi=39\pi$ $cm^{3}$.

Answer:

B. $39\pi$ $cm^{3}$