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The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?
- a)12 cm and 5 cm
- b)24 cm and 10 cm
- c)60 cm and 25 cm
- d)30 cm and 12.5 cm
Correct answer is option 'D'. Can you explain this answer?
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Verified Answer
The diameters of given circles are in the ratio 12 : 5 and the sum of ...
Let the diameters of the circles be a and b cm respectively
⇒ a/b = 12/5
⇒ a2/b2 = 144/25
Area ∝ Diameter2
⇒ a/b = 12/5
⇒ a2/b2 = 144/25
Area ∝ Diameter2
⇒ Sum of area/Area of circle of diameter b = (a2 + b2)/b2 = (144 + 25)/25 = 169/25
⇒ Sum of area of the triangles of the circles is same as the circle with diameter 65 cm
∴ 169/25 = (65)2/b2
∴ b = 25 cm and a = 12/5 × 25 = 60 cm
Radii = 1/2 × Diameter
∴ 169/25 = (65)2/b2
∴ b = 25 cm and a = 12/5 × 25 = 60 cm
Radii = 1/2 × Diameter
∴ Radii of the circle = 60/2 and 25/2 = 30 cm and 12.5 cm
Most Upvoted Answer
The diameters of given circles are in the ratio 12 : 5 and the sum of ...
Given:
- Ratio of diameters of circles: 12 : 5
- Sum of their areas = area of a circle with diameter 65 cm
To find:
Radii of the circles
Solution:
Let the diameters of the circles be 12x and 5x respectively, where x is a constant.
Step 1: Calculate the radii
The radii of the circles are half of their respective diameters.
- Radius of the first circle = (12x)/2 = 6x
- Radius of the second circle = (5x)/2 = 2.5x
Step 2: Calculate the areas
The area of a circle is given by the formula: A = πr^2, where A is the area and r is the radius.
- Area of the first circle = π(6x)^2 = 36πx^2
- Area of the second circle = π(2.5x)^2 = 6.25πx^2
Step 3: Set up the equation
The sum of the areas of the two circles is equal to the area of a circle with diameter 65 cm.
- 36πx^2 + 6.25πx^2 = π(65/2)^2
Simplifying the equation:
- 42.25πx^2 = π(32.5)^2
- 42.25x^2 = 32.5^2
- 42.25x^2 = 1056.25
Step 4: Solve for x
Divide both sides of the equation by 42.25:
- x^2 = 1056.25/42.25
- x^2 = 25
- x = √25
- x = 5
Step 5: Calculate the radii
Substitute the value of x back into the radii formulas:
- Radius of the first circle = 6x = 6(5) = 30 cm
- Radius of the second circle = 2.5x = 2.5(5) = 12.5 cm
Therefore, the radii of the circles are 30 cm and 12.5 cm, which corresponds to option D.
- Ratio of diameters of circles: 12 : 5
- Sum of their areas = area of a circle with diameter 65 cm
To find:
Radii of the circles
Solution:
Let the diameters of the circles be 12x and 5x respectively, where x is a constant.
Step 1: Calculate the radii
The radii of the circles are half of their respective diameters.
- Radius of the first circle = (12x)/2 = 6x
- Radius of the second circle = (5x)/2 = 2.5x
Step 2: Calculate the areas
The area of a circle is given by the formula: A = πr^2, where A is the area and r is the radius.
- Area of the first circle = π(6x)^2 = 36πx^2
- Area of the second circle = π(2.5x)^2 = 6.25πx^2
Step 3: Set up the equation
The sum of the areas of the two circles is equal to the area of a circle with diameter 65 cm.
- 36πx^2 + 6.25πx^2 = π(65/2)^2
Simplifying the equation:
- 42.25πx^2 = π(32.5)^2
- 42.25x^2 = 32.5^2
- 42.25x^2 = 1056.25
Step 4: Solve for x
Divide both sides of the equation by 42.25:
- x^2 = 1056.25/42.25
- x^2 = 25
- x = √25
- x = 5
Step 5: Calculate the radii
Substitute the value of x back into the radii formulas:
- Radius of the first circle = 6x = 6(5) = 30 cm
- Radius of the second circle = 2.5x = 2.5(5) = 12.5 cm
Therefore, the radii of the circles are 30 cm and 12.5 cm, which corresponds to option D.
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The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer?
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The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer?.
The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer? for Defence 2024 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Defence 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer?.
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The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of The diameters of given circles are in the ratio 12 : 5 and the sum of their area is equal to the area of a circle of diameter 65 cm. What are their radii?a)12 cm and 5 cmb)24 cm and 10 cmc)60 cm and 25 cmd)30 cm and 12.5 cmCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
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