Answer :
Area of equilateral triangle = √3/4 a²
49√3 = √3/4 a²
49×4 = a²
a = 7×2
a = 14 cm
Side of equilateral triangle = Side of square ABCD = 14cm
Area of square ABCD = a²
A = 14×14
A = 196 cm²
Now diameter of circle = side of square = 14cm
Radius = D/2 = 7 Cm
There are two half circle in the square = one whole circle
Area of circle = πr²
A = 3.14 ×7 ×7
A = 153.86
Area of yellow region = Area of square - Area of circle
Area of yellow region = 196 - 153.86 = 42.14 Cm²
The area covered by the yellow region (in cm²) in the given diagram is 42cm². The correct option is c.42
From the question, the area of the blue equilateral triangle in the given diagram is 49√3cm².
The area of an equilateral triangle can be calculated by finding half the product of the square of a side (all sides are equal in an equilateral triangle) and sin 60° (each interior angle of an equilateral triangle is 60°)
Hence, the area of the blue triangle is given by
[tex]A = \frac{1}{2}/AB/^{2}sin60\°[/tex]
Where A is the area
Therefore,
[tex]49\sqrt{3} = \frac{1}{2}/AB/^{2}sin60\°\\2 \times 49\sqrt{3} = /AB/^{2} \times \frac{\sqrt{3} }{2}[/tex]
[tex]2 \times2 \times 49\sqrt{3} = /AB/^{2} \times \sqrt{3}[/tex]
[tex]196 \sqrt{3} = /AB/^{2} \times \sqrt{3}[/tex]
∴ [tex]/AB/^{2} = \frac{196 \sqrt{3} }{\sqrt{3} }[/tex]
[tex]/AB/^{2} = 196[/tex]
∴ [tex]/AB/ = \sqrt{196}[/tex]
[tex]/AB/ = 14cm[/tex]
Hence, the length of a side of the square is 14 cm and the diameter of the circle is also 14 cm (since the length of a side of the square cuts the circle into equal halves, hence it is the diameter)
Area covered by the yellow region = Area of the square - ( sum of the areas of the two semicircles)
Sum of the areas of the two semicircles = Area of one circle
Area of a circle is given by the formula
[tex]A = \pi r^{2}[/tex]
But r = d/2
where r is radius and d is diameter
∴ r = 14/2 cm
r = 7cm
Hence
Area of the circle = [tex]\pi \times 7^{2}[/tex]
A = [tex]49\pi[/tex]
Taking π = 22/7
∴ A = [tex]49 \times \frac{22}{7}[/tex]
A = 154cm²
This is the sum of areas of the two semicircles
Now, for the area of the square
Area of the square = /AB/²
∴ Area of the square = 196 cm²
Now, as stated earlier,
Area covered by the yellow region = Area of the square - ( sum of the areas of the two semicircles)
∴ Area covered by the yellow region = 196 cm² - 154cm²
Area covered by the yellow region = 42cm²
Hence, the area covered by the yellow region (in cm²) in the given diagram is 42cm²
The correct option is c.42
Learn more on area of shapes here: https://brainly.com/question/24220654