Rotate the figure below around the straight line A as a rotation axis and make a solid.
Pi is assumed to be 3.14.
(1) Find the area of the cut surface when you cut this solid by the plane including the axis.
(2) Find the volume of this solid.
Answer
(1) 93 cm2
(2) 499.26 cm3
Solution
(1) The area of the cut surface is twice as much as the area of the figure below.
P + Q + R = 2 × 2 +(2+5) ×5 / 2 + 5 ×5 = 4 + 35/2 + 25 = 93/2cm2.
Therefore, an area to find is 93/2 × 2 = 93cm2.
(2) The volume of the solid by the rotation of P is 2 × 2 × 3.14 × 2 = 8 × 3.14.
The volume of the solid by the rotation of R is 5 × 5 × 3.14 × 5 = 125 × 3.14.
△ BCD and △ BEF are homothetic and the homothetic ratio is 5 : 2.
Because BC : BE = 5 : 2, BE = 2cm × 2/(5-2) = 4/3 cm.
The volume of the solid by the rotation of Q is 5 × 5 × 3.14 × (4/3+2) × 1/3 - 2 × 2 × 3.14 × 4/3 × 1/3 = 250/9 × 3.14 - 16/9 × 3.14 = 26 × 3.14.
Therefore the volume to find is 8 × 3.14 + 125 × 3.14 + 26 × 3.14 = 159 × 3.14 = 499.26 cm3.
