(a) 270 cm3
The base of the prism is a pentagon as shown.
If the longest side is 6 cm, then the other sides are 3 cm, 3√2 cm, 3√2 cm, 3 cm
Since ABC is an isosceles triangle, AF ⊥ BC and ∠AFB = 90º.
∴ BF = FC = 3 cm
In ΔABF, AF = \(\sqrt{(3\sqrt2)^2-3^2}=\sqrt{18-9}=\sqrt9=3.\)
∴ Total area of the base = Area of ΔABC + Area of rect. BCDE
= \(\big(\frac12\times6\times3+6\times3\big)\) cm2 = (9+18) cm2 = 27 cm2.
∴ Volume of prism = Area of base × height
= (27 × 10) cm3 = 270 cm3.