The answer is 176π cm².
Given shape is a combination of 3 semi-circles.
And we need to find the area of the colored region.
The diameter of the biggest semi-circle is 32 cm.
Let d₁, and r₁ denote the diameter and radius of the biggest semi-circle respectively.
⇒ d₁ = 32 cm and r₁ = 32/2 = 16
The diameter of the smallest semi-circle is 4cm.
Let d₂, and r₂ denote the diameter and radius of the smallest semi-circle.
⇒ d₂ = 4 cm and r₂ = 4/2 = 2
Let d₃, and r₃ denote the diameter and radius of the middle semi-circle.
From the diagram, the diameter of the middle semi-circle is,
d₃ = d₁ - (d₂ + 8) = 32 -(4 + 8) = 32 - 12 = 20
⇒ d₃ = 20 cm and r₂ = 20/2 = 10
The area of a semi-circle is given by the formula πr²/2, where r is the radius.
Let A₁ denote the area of the biggest semi-circle.
Then A₁ = π(r₁)²/2 = π × (16)² / 2 = 256π/2 = 128π cm²
Let A₂ denote the area of the smallest semi-circle.
Then A₂ = π(r₂)²/2 = π × (2)² / 2 = 4π/2 = 2π cm²
Let A₃ denote the area of the middle semi-circle.
Then A₃ = π(r₃)²/2 = π × (10)² / 2 = 100π/2 = 50π cm²
Therefore, the area of the colored region is A₁ + A₃ - A₂
= 128π + 50π - 2π
= 176π cm²
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