The surface area and volume of a rectangular block in which the length is 1 centimetre more than the width and the height is 1 centimetre more than the length.

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Solution

Let the width of the rectangular block be x cm.

∴ Length of the rectangular block = (x + 1) cm

Height of the rectangular block = (x + 1 + 1) cm = (x + 2) cm

Volume of the rectangular block = Length × Width × Height

= (x + 1) × x × (x + 2) cm³

= (x² + x) × (x + 2) cm³

= {x²(x + 2) + x(x + 2)} cm³

= (x³ + 2x² + x² + 2x) cm³

= {x³ + (2 + 1) x² + 2x} cm³

= (x³ + 3x² + 2x) cm³

Surface area of the rectangular block = 2(lb + bh + hl)

= 2{(x + 1)x + x(x + 2) + (x + 2)(x + 1)} cm²

= 2{x² + x + x² + 2x + x² + x + 2x + 2} cm²

= 2{(1 + 1 + 1)x² + (1 + 2 + 1 + 2)x + 2} cm²

= 2(3x² + 6x + 2) cm²

= (6x² + 12x + 4) cm²

Yes, the algebraic expressions for the volume and surface area of the rectangular block are polynomials because the exponents of the variables in the expressions are natural numbers.

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