The cross-section of a 6 m long piece of metal is shown in the figure. Calculate :

(i) the area of the cross-section

(ii) the volume of the piece of metal in cubic centimeters.

From figure,

AD = BC = 8 cm

AB = CD = 6.5 cm

Triangle sides = AE = 5 cm & DE = 5 cm

Triangle base (AD) = 8 cm.

Length of metal piece = 6 m = 600 cm.

(i) Area of cross section :

Calculating the area of rectangle ABCD,

Area of rectangle ABCD = length × breadth

= 8 × 6.5

= 52 cm².

EAD is an isosceles triangle,

In an isosceles triangle, the perpendicular drawn from common vertex to the base, bisects the base.

AF = FD = $\dfrac{AD}{2} = \dfrac{8}{2}$ = 4 cm.

By using pythagoras theorem for the triangle EFD,

⇒ Hypotenuse² = Base² + Height²

⇒ ED² = FD² + EF²

⇒ 5² = 4² + EF²

⇒ 25 = 16 + EF²

⇒ EF² = 25 - 16

⇒ EF² = 9

⇒ EF = $\sqrt{9}$

⇒ EF = 3 cm.

∴ Height (EF) = 3 cm

Area of △ EAD = $\dfrac{1}{2}$ × Base × Height

= $\dfrac{1}{2}$ × AD × EF

= $\dfrac{1}{2}$ × 8 × 3

= 4 × 3

= 12 cm².

Total area = Area of rectangle ABCD + Area of triangle EAD

= 52 + 12 = 64 cm².

Hence, area of cross-section = 64 cm².

(ii) Volume of metal:

Volume of metal = Area of cross-section × Length of the metal piece

= 64 × 600

= 38400 cm³.

Hence, volume of metal = 38400 cm³.