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Three toys are in the shape of the cylinder, hemisphere, and cone . The three toys have the same base. The height of each toy is 2 √2. What is the ratio of the total surface areas of the cylinder, hemisphere and cone respectively ?
- a)4 : 3 : √2 +1
- b)4 : 3 : 2 + √2
- c)4 : 3 : 2 √2
- d)2 : 1 : √2 + 1
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Three toys are in the shape of the cylinder, hemisphere, and cone . T...
Height of each toy = 2√2
since height of hemisphere = radius of hemisphere
= radius of each toy = 2√2
slant height of cone(l)
ratio = T.S.A of cylinder: T.S.A of the hemisphere : T.S.A of cone
Most Upvoted Answer
Three toys are in the shape of the cylinder, hemisphere, and cone . T...
The Ratio of Total Surface Areas of the Cylinder, Hemisphere, and Cone
To find the ratio of the total surface areas of the cylinder, hemisphere, and cone, we need to calculate the individual surface areas of each shape and then compare them.
Given:
- The three toys have the same base.
- The height of each toy is 2√2.
Surface Area of a Cylinder:
The surface area of a cylinder is given by the formula:
A = 2πrh + πr^2, where r is the radius and h is the height.
Since the three toys have the same base, their radii will also be the same. Let's assume the radius of each toy is 'r'.
The height of each toy is given as 2√2, so the surface area of the cylinder will be:
A_cylinder = 2πr(2√2) + πr^2
= 4πr√2 + πr^2
Surface Area of a Hemisphere:
The surface area of a hemisphere is given by the formula:
A = 2πr^2, where r is the radius.
Since the three toys have the same base, their radii will also be the same. Let's assume the radius of each toy is 'r'.
The surface area of the hemisphere will be:
A_hemisphere = 2πr^2
Surface Area of a Cone:
The surface area of a cone is given by the formula:
A = πrl + πr^2, where r is the radius and l is the slant height.
Since the three toys have the same base, their radii will also be the same. Let's assume the radius of each toy is 'r'.
The height of each toy is given as 2√2, so the slant height of the cone will be:
l = √(r^2 + (2√2)^2)
= √(r^2 + 8)
The surface area of the cone will be:
A_cone = πr(√(r^2 + 8)) + πr^2
= πr√(r^2 + 8) + πr^2
Calculating the Ratio:
To calculate the ratio of the total surface areas, we need to add up the surface areas of the cylinder, hemisphere, and cone and then compare them.
Total surface area = A_cylinder + A_hemisphere + A_cone
Plugging in the formulas we derived earlier, we get:
Total surface area = (4πr√2 + πr^2) + (2πr^2) + (πr√(r^2 + 8) + πr^2)
Now, simplify the expression and factor out common terms:
Total surface area = 4πr√2 + 2πr^2 + πr√(r^2 + 8)
Now, we can calculate the ratio of the total surface areas by dividing each term by the total surface area:
Ratio = (4πr√2) / (4πr√2 + 2πr^2 + πr√(r^2 + 8))
To find the ratio of the total surface areas of the cylinder, hemisphere, and cone, we need to calculate the individual surface areas of each shape and then compare them.
Given:
- The three toys have the same base.
- The height of each toy is 2√2.
Surface Area of a Cylinder:
The surface area of a cylinder is given by the formula:
A = 2πrh + πr^2, where r is the radius and h is the height.
Since the three toys have the same base, their radii will also be the same. Let's assume the radius of each toy is 'r'.
The height of each toy is given as 2√2, so the surface area of the cylinder will be:
A_cylinder = 2πr(2√2) + πr^2
= 4πr√2 + πr^2
Surface Area of a Hemisphere:
The surface area of a hemisphere is given by the formula:
A = 2πr^2, where r is the radius.
Since the three toys have the same base, their radii will also be the same. Let's assume the radius of each toy is 'r'.
The surface area of the hemisphere will be:
A_hemisphere = 2πr^2
Surface Area of a Cone:
The surface area of a cone is given by the formula:
A = πrl + πr^2, where r is the radius and l is the slant height.
Since the three toys have the same base, their radii will also be the same. Let's assume the radius of each toy is 'r'.
The height of each toy is given as 2√2, so the slant height of the cone will be:
l = √(r^2 + (2√2)^2)
= √(r^2 + 8)
The surface area of the cone will be:
A_cone = πr(√(r^2 + 8)) + πr^2
= πr√(r^2 + 8) + πr^2
Calculating the Ratio:
To calculate the ratio of the total surface areas, we need to add up the surface areas of the cylinder, hemisphere, and cone and then compare them.
Total surface area = A_cylinder + A_hemisphere + A_cone
Plugging in the formulas we derived earlier, we get:
Total surface area = (4πr√2 + πr^2) + (2πr^2) + (πr√(r^2 + 8) + πr^2)
Now, simplify the expression and factor out common terms:
Total surface area = 4πr√2 + 2πr^2 + πr√(r^2 + 8)
Now, we can calculate the ratio of the total surface areas by dividing each term by the total surface area:
Ratio = (4πr√2) / (4πr√2 + 2πr^2 + πr√(r^2 + 8))
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Three toys are in the shape of the cylinder, hemisphere, and cone . The three toys have the same base. The height of each toy is 2 √2. What is the ratio of the total surface areas of the cylinder, hemisphere and cone respectively ?a)4 : 3 : √2 +1b)4 : 3 : 2 + √2c)4 : 3 : 2 √2d)2 : 1 : √2 + 1Correct answer is option 'A'. Can you explain this answer?
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