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From a circular sheet of paper with radius 20 cm, four circles of radius 5 cm each are cut out. What is the ratio of the area of the uncut portion to that of the cut portion?
- a)1 : 3
- b)4 : 1
- c)3 : 1
- d)4 : 3
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
From a circular sheet of paper with radius 20 cm, four circles of rad...
To solve this problem, we need to calculate the areas of the uncut portion and the cut portion separately and then find their ratio.
Let's start by finding the area of the uncut portion:
- The circular sheet of paper has a radius of 20 cm, so its area is given by the formula πr^2, where r is the radius.
- The area of the uncut portion is equal to the area of the large circle minus the area of the four smaller circles.
- The area of the large circle is π(20 cm)^2.
- The area of each smaller circle is π(5 cm)^2.
- Therefore, the area of the uncut portion is π(20 cm)^2 - 4π(5 cm)^2.
Next, let's find the area of the cut portion:
- The cut portion consists of four circles, each with a radius of 5 cm.
- The total area of the cut portion is equal to the sum of the areas of the four smaller circles, which is 4π(5 cm)^2.
Now, let's simplify the expressions for the areas:
- The area of the uncut portion is π(20 cm)^2 - 4π(5 cm)^2 = π(400 cm^2) - 4π(25 cm^2) = π(400 cm^2 - 100 cm^2) = π(300 cm^2).
- The area of the cut portion is 4π(5 cm)^2 = 4π(25 cm^2) = π(100 cm^2).
Finally, let's find the ratio of the areas:
- The ratio of the area of the uncut portion to that of the cut portion is (π(300 cm^2))/(π(100 cm^2)).
- The π cancels out, leaving us with the ratio 300 cm^2/100 cm^2.
- Simplifying this ratio gives us 3/1.
Therefore, the ratio of the area of the uncut portion to that of the cut portion is 3:1, which corresponds to option C.
Let's start by finding the area of the uncut portion:
- The circular sheet of paper has a radius of 20 cm, so its area is given by the formula πr^2, where r is the radius.
- The area of the uncut portion is equal to the area of the large circle minus the area of the four smaller circles.
- The area of the large circle is π(20 cm)^2.
- The area of each smaller circle is π(5 cm)^2.
- Therefore, the area of the uncut portion is π(20 cm)^2 - 4π(5 cm)^2.
Next, let's find the area of the cut portion:
- The cut portion consists of four circles, each with a radius of 5 cm.
- The total area of the cut portion is equal to the sum of the areas of the four smaller circles, which is 4π(5 cm)^2.
Now, let's simplify the expressions for the areas:
- The area of the uncut portion is π(20 cm)^2 - 4π(5 cm)^2 = π(400 cm^2) - 4π(25 cm^2) = π(400 cm^2 - 100 cm^2) = π(300 cm^2).
- The area of the cut portion is 4π(5 cm)^2 = 4π(25 cm^2) = π(100 cm^2).
Finally, let's find the ratio of the areas:
- The ratio of the area of the uncut portion to that of the cut portion is (π(300 cm^2))/(π(100 cm^2)).
- The π cancels out, leaving us with the ratio 300 cm^2/100 cm^2.
- Simplifying this ratio gives us 3/1.
Therefore, the ratio of the area of the uncut portion to that of the cut portion is 3:1, which corresponds to option C.
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Community Answer
From a circular sheet of paper with radius 20 cm, four circles of rad...
Area of four small circles = 4 × π × 25 cm2 = 100π cm2
Area of the big circle = π × 20 × 20 cm2 = 400π cm2
Area of the uncut portion = (400π - 100π) cm2 = 300π cm2
Ratio of uncut to cut area = 300π : 100π = 3 : 1
Hence, answer option 3 is correct.
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From a circular sheet of paper with radius 20 cm, four circles of radius 5 cm each are cut out. What is the ratio of the area of the uncut portion to that of the cut portion?a)1 : 3b)4 : 1c)3 : 1d)4 : 3Correct answer is option 'C'. Can you explain this answer?
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