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A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.?
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A crescent is formed by two circles which touch at A. C is the centre ...
Let x = DC, C ' be the center of the smaller circle, r be the radius of the smaller circle, and R be the radius of the larger circle. We have
BC = BD+DC = 9+x
and
AC = AD-CD = 2r-x.
Since
R = BC = AC,
(1) R = 9+x = 2r-x,
(2) r = (2x+9)/2.
In the right triangle C 'CE,
CE = CF-EF = R-5 = x+9-5 = x+4,
C 'C = C 'D-CD = r-x,
and
C 'E = r.
By the Pythagorean theorem,
(x+4)^2+(r-x)^2 = r^2,
r^2-(r-x)^2 = (x+4)^2,
(r-r+x)(r+r-x) = (x+4)^2,
x(2r-x) = (x+4)^2.
Using (1) in the last equation, we have
x(9+x) = (x+4)^2,
x = 16.
It follows that, by (2), r = (2x+9)/2 = 41/2 and by (1), R = 9+x = 25. Hence the area of the crescent is pi*(R^2-r^2) = pi*(R-r)(R+r) = pi*(9/2)*(91/2) = 819*pi/4 = 643.5 cm^2
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A crescent is formed by two circles which touch at A. C is the centre ...
Understanding the Crescent Shape
The crescent is formed by the intersection of two circles, creating a shaded area between them. To calculate the area of the shaded region, we need to determine the areas of both circles involved.
Dimensions of the Crescent
- The width of the crescent at BD is 9 cm.
- The width at EF is 5 cm.
Radii of the Circles
- Let R be the radius of the larger circle.
- Let r be the radius of the smaller circle.
The width of the crescent represents the difference in radii at the points where the circles intersect.
Calculating the Radii
- The larger circle's radius can be derived from the width at BD:
R = (Width at BD) / 2 = 9 cm / 2 = 4.5 cm.
- The smaller circle's radius can be derived from the width at EF:
r = (Width at EF) / 2 = 5 cm / 2 = 2.5 cm.
Area Calculation
- The area of the larger circle (A1) is given by the formula:
A1 = πR^2 = π(4.5)^2 = 20.25π cm².
- The area of the smaller circle (A2) is given by the formula:
A2 = πr^2 = π(2.5)^2 = 6.25π cm².
Area of the Shaded Region
- The area of the shaded region (A) is calculated by subtracting the area of the smaller circle from the area of the larger circle:
A = A1 - A2 = 20.25π - 6.25π = 14π cm².
Thus, the area of the shaded region is 14π cm² or approximately 43.98 cm² when calculated numerically.
The crescent is formed by the intersection of two circles, creating a shaded area between them. To calculate the area of the shaded region, we need to determine the areas of both circles involved.
Dimensions of the Crescent
- The width of the crescent at BD is 9 cm.
- The width at EF is 5 cm.
Radii of the Circles
- Let R be the radius of the larger circle.
- Let r be the radius of the smaller circle.
The width of the crescent represents the difference in radii at the points where the circles intersect.
Calculating the Radii
- The larger circle's radius can be derived from the width at BD:
R = (Width at BD) / 2 = 9 cm / 2 = 4.5 cm.
- The smaller circle's radius can be derived from the width at EF:
r = (Width at EF) / 2 = 5 cm / 2 = 2.5 cm.
Area Calculation
- The area of the larger circle (A1) is given by the formula:
A1 = πR^2 = π(4.5)^2 = 20.25π cm².
- The area of the smaller circle (A2) is given by the formula:
A2 = πr^2 = π(2.5)^2 = 6.25π cm².
Area of the Shaded Region
- The area of the shaded region (A) is calculated by subtracting the area of the smaller circle from the area of the larger circle:
A = A1 - A2 = 20.25π - 6.25π = 14π cm².
Thus, the area of the shaded region is 14π cm² or approximately 43.98 cm² when calculated numerically.
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A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.?
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A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.?.
A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A crescent is formed by two circles which touch at A. C is the centre of the larger circle. The width of the crescent at BD is 9 cm and at EF is 5 cm. Find area of the shaded region.?.
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