Question

A rectangular carpet has an area 60 m². If the sum of its diagonal and longer side is equal to 5 times the shorter side, find the breadth of the carpet.

Answer 1

Given:

Area = 60 m².

Let breadth (shorter side) = b (m), length (longer side) = l (m), diagonal = d (m).

Area relation: l × b = 60.

Diagonal (Pythagoras): `d = sqrt(1^2 + b^2)`.

Condition: d + l = 5b.

Step-wise calculation:

1. From area:

`l = 60/b`

2. Substitute into the condition:

`sqrt(l^2 + b^2) + l = 5b`

⇒ `sqrt((60/b)^2 + b^2) + 60/b = 5b`

3. Isolate the square root:

`sqrt(3600/b^2 + b^2) = 5b - 60/b`

4. Square both sides:

`3600/b^2 + b^2 = (5b - 60/b)^2`

= `25b^2 - 600 + 3600/b^2`

Note `3600/b^2` cancels from both sides.

5. This leaves: b² = 25b² – 600

⇒ 24b² = 600

⇒ b² = `600/24`

⇒ b² = 25 

⇒ b = 5 (reject negative length).

6. Find the other dimensions to check consistency:

`l = 60/b` 

= `60/5`

= 12 m

`d = sqrt(l^2 + b^2)` 

= `sqrt(144 + 25)`

= `sqrt(169)`

= 13 m

Check: d + l = 13 + 12 

= 25 = 5 × 5

= 5b

Breadth of the carpet = 5 m.