The length of an aluminium strip is 40 cm. If the lengths in cm are measured in natural numbers, write the measurement of all the possible rectangular frames which can be made out of it.
Possible dimensions (Length × Breadth) = (19×1), (18×2), (17×3), (16×4), (15×5), (14×6), (13×7), (12×8), (11×9), (10×10)
Step 1: Understand the given strip length
The strip makes the boundary of a rectangle. Boundary means perimeter.
Step 2: Recall the perimeter formula
For a rectangle: ( P = 2(L + B) )
Step 3: Substitute the given perimeter
Given ( P = 40 ) cm.
So, ( 40 = 2(L + B) )
Step 4: Simplify
Divide both sides by 2:
( L + B = 20 )
Step 5: Use natural numbers only
We are told lengths are measured in natural numbers (1, 2, 3, …).
So ( L ) and ( B ) must be whole positive numbers.
Step 6: Avoid duplicates by taking ( L ge B )
A frame of ( (L, B) ) is the same as ( (B, L) ).
So we list pairs with ( L ge B ) and ( L + B = 20 ).
Step 7: List all pairs that add to 20
Step 8: Write the dimensions (Length × Breadth)
((19 imes 1), (18 imes 2), (17 imes 3), (16 imes 4), (15 imes 5), (14 imes 6), (13 imes 7), (12 imes 8), (11 imes 9), (10 imes 10))
Check: Each pair adds to 20 (so the perimeter is 40), and both numbers are natural numbers.
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