Question

A solid copper piece has the shape shown in figure. (All measurements are in cm). The face ABCDEFA is the uniform cross-section. Assume that the angles at A, B, C, D, E and F are right angles.

1. Calculate the area of the uniform cross-section.
2. Calculate the volume of the above piece.

Answer 1

Given: Cross-section ABCDEFA is a right-angled rectilinear shape with the step dimensions: vertical AB = 4 cm, horizontal BC = 5 cm, vertical CD = 6 cm, horizontal DE = 3 cm. The prism length = 22 cm.

Step-wise calculation:

1. Place coordinates to match the vertices (x to right, y down):

A(0, 0), B(0, 4), C(5, 4), D(5, 10), E(8, 10), F(8, 0)

2. Enclose the polygon area by decomposition:

Bounding rectangle width = 8 cm (5 + 3), height = 10 cm (4 + 6).

Area of bounding rectangle = 8 × 10 = 80 cm².

Missing rectangular corner from x = 0 to 5, y = 4 to 10 has area = 5 × 6 = 30 cm².

Area of cross-section = 80 – 30 = 50 cm². (Alternative check by shoelace gives the same 50 cm².)

3. Volume of the solid = (Area of cross-section) × (Length of prism)

Volume = 50 cm² × 22 cm

= 1100 cm³.

Area of the uniform cross-section = 50 cm².

Volume of the solid copper piece = 1100 cm³.