Suppose the length of a rectangular box is 7 cm and breadth is 11 cm. Find the volume if the height is (i) 5 cm, (ii) 9 cm, (iii) 13 cm. Find the linear pattern representing the volume of the rectangular box.
Step-by-step solution
Idea: Volume = length × breadth × height. The base area 7 × 11 = 77 cm2 is fixed, so the volume is 77 times the height.
- Base area = 7 × 11 = 77 cm2, so V = 77 × height.
- (i) Height 5 cm: V = 77 × 5 = 385 cm3.½ mark
- (ii) Height 9 cm: V = 77 × 9 = 693 cm3.½ mark
- (iii) Height 13 cm: V = 77 × 13 = 1001 cm3.½ mark
- For height h cm: V = 77h cm3, a linear expression in h.1 mark
- Heights 5, 9, 13 go up by 4; volumes 385, 693, 1001 go up by 77 × 4 = 308 each time. The constant difference shows a linear pattern.½ mark
Check: 693 − 385 = 308 and 1001 − 693 = 308 ✓.
Answer to write in the exam
Volume = l × b × h = 7 × 11 × h = 77h
(i) 77 × 5 = 385 cm3
(ii) 77 × 9 = 693 cm3
(iii) 77 × 13 = 1001 cm3
∴ Linear pattern: V = 77h (volumes increase by 308 cm3 for each 4 cm of height)
Common mistakes that cost marks
- Adding the dimensions (7 + 11 + 5 = 23) instead of multiplying them.
- Working out 77 × 13 as 911 or 1011. 77 × 13 = 770 + 231 = 1001.
- Writing the volume in cm2 instead of cm3.
How this can come in the exam
A box has length 6 cm and breadth 5 cm. If its height increases by 2 cm, its volume increases by
- 11 cm3
- 30 cm3
- 60 cm3
- 22 cm3
Show answer
(C) 60 cm3
V = 30h, so 2 cm more adds 30 × 2 = 60 cm3.
A tank has a base 40 cm by 25 cm. Write the volume of water when the water is h cm deep, and find it when h = 18.
Show answer
V = 40 × 25 × h = 1000h cm3 (1 mark). At h = 18: 18000 cm3 (1 mark).Try one yourself
A box is 8 cm long and 6 cm wide. Find its volume for heights 3, 5 and 7 cm, and write the pattern.
Show answer
144, 240, 336 cm3; V = 48h (up by 96 cm3 for each 2 cm).
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