Question

In the following figure, the boundary of the shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm, find
(i)the length of the boundary.
(ii)the area of the shaded region.1​

Answer 1

Answer:

The area of the shaded region is 86.625 cm² .

Step-by-step explanation:

Given :  

The shaded region consists of four semi-circular arcs, the smallest two being equal. If the diameter of the largest is 14 cm and of the smallest is 3.5 cm.

Length of the boundary = boundary of a semicircle with diameter 14 cm + boundary of a semicircle with diameter 7 cm + 2×   boundary of semicircle with diameter 3.5 cm

Length of the boundary = π(14/2) + π(7/2) + 2× π(3.5/2)

= 7π + 3.5π + 3.5π

= 7π + 7π  

= 14π  

= 14 × 22/7  

= 2 × 22

= 44 cm

Length of the boundary = 44 cm

Area of the shaded region ,A = Area of semicircle with AB as diameter - Area of the semicircle with radius AE - Area of the semicircle with radius BC + Area of semicircle with diameter 7 cm

A = ½ × π(14/2)² - ½ × π(3.5/2)² - ½ × π(3.5/2)² + ½ × π(7/2)²

A =1/2π [7² - 1.75² - 1.75² + 3.5²]

A = ½ π[49 - 3.0625 - 3.0625 + 12.25]

A = ½ π[49 - 6.125 + 12.25]

A = ½ π [42.875 + 12.25]

A = ½ π [55.125]

A = ½ × 22/7 × 55.125

A = 11 × 7.875

A = 86.625 cm²

Area of the shaded region = 86.625 cm²

Hence, the area of the shaded region is 86.625 cm² .

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Answer 2

$\huge\underline\mathcal{Solution}$

Given that , the radius of largest semicircle (R ) = 7 cm .

Radius of the middle semicircle = 3.5 cm

$= \frac{7}{2} cm$

Radius of the smallest semicircle

$= \frac{3.5}{2}$

$= \frac{7 \div 2}{2}$

$= \frac{7}{4} cm$

(i) Length of boundary = boundary of a semicircle with radius 7 cm + boundary of a semi circle with radius 7/2 cm + 2 × boundary of semi circle with radius 7/4 cm .

$=7 \pi \: + \frac{7}{2} \pi \: + 2 \times \frac{7}{4} \pi$

$= 7\pi(1 + \frac{1}{2} + \frac{1}{2} )$

$= 14\pi$

(ii) Area of shaded region = Area of largest semi circle + area of middle semi circle + 2 × area od smallest semi circle

$= \frac{1}{2} \pi {7}^{2} + \frac{1}{2} \pi( { \frac{7}{2} }^{2} ) - 2 \times \frac{1}{2} \pi( { \frac{7}{4} )}^{2}$

$= \frac{1}{2} \pi \times {7}^{2} + \frac{1}{2} \pi \times \frac{ {7}^{2} }{ {2}^{2} } - \pi \times \frac{ {7}^{2} }{ {4}^{2} }$

$= {7}^{2} \pi( \frac{1}{2} + \frac{1}{2} \times \frac{1}{4} - \frac{1}{16} )$

$= 49\pi( \frac{1}{2} + \frac{1}{8} - \frac{1}{16} )$

$= 49 \times \frac{22}{7} ( \frac{8 + 2 - 1}{16} )$

$= \frac{22}{7} \times 7 \times 7 \times \frac{9}{16}$

$= 22 \times 7 \times \frac{9}{16}$

$= 86.625 {cm}^{2}$

Hence , the area of the shaded region is 86.625cm^2.