Area and Volume as Rules പരപ്പളവും വ്യാപ്തവും
On the last page the perimeter of a stretched rectangle became the rule p(x) = 4x + 10. Now we do the same for its area, and then for the volume of a stretched block. This time squares x² and cubes x³ appear – and the rules start to look like real polynomials.
- x² = x × x and x³ = x × x × x.
- (x + a)(x + b) = x² + (a + b)x + ab – the multiplication identities (ഗുണനസമവാക്യങ്ങൾ, chapter 4).
- p(3) means “the value of p when x is 3”.
1. The area of the stretched rectangle
Back to the 3 cm × 2 cm rectangle. Instead of trying one added length after another, the textbook starts straight away with a general added length of x cm and cuts the new rectangle into pieces.
If every side of the 3 cm × 2 cm rectangle is lengthened by x cm and the new area (പരപ്പളവ്) is a(x) square centimetres, then
In algebra we usually write the letters first: x² + 5x + 6 rather than 6 + 5x + x² – the same thing.
Four pieces: the old 6, two strips 3x and 2x, and a little square x². The strips together are 5x.
2. Using the rule: a(1), a(1½), a(2)
- If all sides are lengthened by 1 cm, the area is 12 square centimetres.
- If all sides are lengthened by 1½ cm, the area is 15¾ square centimetres.
- If all sides are lengthened by 2 cm, the area is 20 square centimetres.
The area does not grow by the same amount each time: a(1) = 12, a(2) = 20, a(3) = 30, a(4) = 42 – the jumps are 8, 10, 12. That is the x² piece at work. The perimeter p(x) = 4x + 10 grew by exactly 4 each time; the area does not.
3. Stretching a block: volume
Now a rectangular block (ചതുരക്കട്ട) with edges 1 cm, 2 cm and 3 cm. Every edge is lengthened by the same x cm. What happens to the volume (വ്യാപ്തം)?
The new edges are x + 1, x + 2 and x + 3, so the new volume is (x + 1)(x + 2)(x + 3). To write it out as a sum we multiply step by step.
If every edge of a block with edges 1 cm, 2 cm and 3 cm is lengthened by x cm, and the volume of the bigger block is v(x) cubic centimetres, then
To multiply two sums, multiply each term of the first by each term of the second, and add (ആദ്യത്തെ തുകയിലെ സംഖ്യകളോരോന്നിനെയും, രണ്ടാമത്തെ തുകയിലെ ഓരോന്നുകൊണ്ടും ഗുണിച്ച്, കൂട്ടണം).
x = 0 means nothing was added: v(0) = 6 = 1 × 2 × 3 ✓. x = 1: the block is 2 × 3 × 4 = 24, and 1 + 6 + 11 + 6 = 24 ✓. Checking with a simple number catches most multiplication slips.
A tailor adds the same seam allowance x to every side of a 30 cm × 20 cm piece: cloth needed (30 + 2x)(20 + 2x) = 4x² + 100x + 600. A baker's cake tin grows by x cm each way: the batter needed is a volume rule like v(x). Shopkeepers, builders and packers use such rules to see quickly how cost changes with size.
4. Check yourself
- Area of the stretched 3 × 2 rectangle: a(x) = (x + 3)(x + 2) = x² + 5x + 6; a(1) = 12, a(1½) = 15¾, a(2) = 20.
- Volume of the stretched 1 × 2 × 3 block: v(x) = (x + 1)(x + 2)(x + 3) = x³ + 6x² + 11x + 6.
- To multiply sums: each term by each term, then collect like terms.
- Perimeter rules have x; area rules have x²; volume rules have x³.
Next: Exercise 1 – Solutions ▶