Rectangle with area 60 — diagonal plus longer side equals 5 times shorter side: A rectangular carpet has an area of 60 sq m. The diagonal and the longer side together equal 5 times the shorter side. Find the length of the carpet (the longer side).

Introduction / Context:This problem blends area, Pythagoras, and a linear relation between diagonal and sides. It is a classic rectangle system intended to test simultaneous reasoning steps.

Given Data / Assumptions:

• Let shorter side = a, longer side = b
• Area: a * b = 60
• Diagonal: d = sqrt(a^2 + b^2)
• Relation: d + b = 5a

Concept / Approach:Use b = 60 / a from area, then substitute into d + b = 5a with d = sqrt(a^2 + (60/a)^2). Solve for a, then compute b. Favor factorable integer triples if they appear.

Step-by-Step Solution:Try a = 5 → b = 60/5 = 12d = sqrt(5^2 + 12^2) = sqrt(25 + 144) = 13Check relation: d + b = 13 + 12 = 25 = 5a (with a = 5) ✓Hence longer side = b = 12 m

Verification / Alternative check:Area 5 * 12 = 60 ✓; Pythagorean triple (5, 12, 13) matches and satisfies the linear condition.

Why Other Options Are Wrong:5 m is the shorter side; 13 m is the diagonal; 14.5 m does not satisfy area and diagonal constraints simultaneously.

Common Pitfalls:Mixing up which side is longer or treating d + a = 5b by mistake.

Final Answer:12 m