Two small circles tangent to each other are inscribed in a rectangle. A bigger circle is circumscribed about the rectangle. If the area of the shaded region is 10 square inches, find the exact area of the circumscribed circle.

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jaewonlee0217fyv · Accepted Answer

Step 1Let the diameter of each small identical circle be denoted as d.The length of rectangle is equal to sum of diameters since the circles are tangent to one another. The width of rectangle equals diameter since the rectangle inscribes the circles.Step 2The length of rectangle will be d+d=2d. The width will be equal to d.Calculate the area of rectangle.Area of rectangle=length×width=2d×d=2d2Calculate area of two identical circles.Area of two circles=2×(π×d24)=πd22The difference in their areas equal 10 square inches. Use this relation to calculate value of d2.2d2−πd22=10d2=204−πThe diagonal of rectangle will be diameter of bigger circle since it is a chord which passes through the center of bigger circle denoted by the point of contact of the two small circles.Apply Pythagoras theorem to the right angled triangle formed by the diagonal of rectangle, its width and its length. Let the diameter of bigger circle be denoted as d&#039;.d′2=d2+(2d)2=5d2Substitute the value of d2d′2=5d2=5×204−π=1004−π The area of bigger circle will be equal to πd′24.πd′24=π4×1004−π=25π4−π=25×3.144−3.14=91.28∈2Hence the area of area of circumscribed circle is 91.28∈2.

chung001vba · Accepted Answer

Step 1Let the radii of the small incscribed circles be r. The area of a circle with a radius r is Aπr2. Let&#039;s redraw the figure again with the dimensions.The area of a rectangle is length×width. The area of the shaded region is equal to the area of the rectangle minus the area of inscribed circles. We have given the area of the shaded region is 10 square inches.Shaded area=area(◻ABCD)−2×area (one small circle)10=(4r)(2r)−2πr210=2r2(4−π)r2=54−πStep 2Again redraw the figure and find the diameter of circumscribed circle using the Pythagorean theorem in the right triangle ACD.AD2=AC2+CD2=(2r)2+(4r)2=4r2+16r2=20r2AD=25rStep 3Since the radius of a circle is half of the diameter. Therefore, R=5r, where R is the radius of the circumscribed circle. Now, find the area of the circumscribed circle.Area=πR2=π(5r)2=25π4−π square inchesHence, the area of the circumscribed circle is 25π4−π square inches.

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