This is a famous question. Just remember whenever you are forming a circle and then a square,
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the side of that square is given by, a = 1.6*r (approx.), where r = radius of the circle
I have written approx. because the actual formula is 1.57*r, but it will make the calculations a bit lengthy. So, just find 1.6*r and the answer will be little less than that. Like here
Side = 1.6*84 = 134.4, so the answer is 132 cm
Answer: (A)
When the wire is bent in the form of a circle of radius 84cm, that means the circumference (or the length of the wire) of the circle is 2*π*84 = 44*12 cm
Now this wire forms a square of (let’s say) side ‘a’
Then, 4a (perimeter of the square) = 44*12
Hence a = 132 cm
If the given rectangular sheet of paper (length =l, breadth = b) is rolled across its length to form a cylinder, having a height b, then volume of cylinder = (l*l*b)/4π
If rolled across its breadth, then = (b*b*l)/4π
In this question the sheet is rolled along its length, so volume = (l*l*b)/4π = 12*12*5/(4* π)
Volume = 180/ π cm^{3}
Answer: (C)
In this question, the ratio of surface areas is given and they are asking the ratio of volumes. The word “sphere” is useless here. In such questions, just imagine area as A^{2} and volume as A^{3}. Now A^{3} is given and you have to find A^{3}. How will you do it? Simple, first take the square-root of A^{2} to convert it into A, and then take the cube of A to find A^{3}.
So for solving this question, we just have to take the square-root of 4:9. The ratio will become 2:3. Then take the cube of 2:3. Hence the answer is 8:27
Answer: (C)
Here again ratio of areas is given, that means A^{2} is given, and we have to find A. So 4:9 will become 2:3
Answer: (A)
Diameter and perimeter are directly proportional, P = D*π, where P is the perimeter and D is the diameter.
Hence a 75% increase in diameter means a 75% increase in perimeter
Answer: (D)
The area of base and the volume of a cone are directly proportional V = A * h/3, where V = volume and h = height of the cone
Hence a 100% increase in the area of the base would mean a 100% increase in the volume
Answer: (B)
A is increased by 50% hence A^{2 }(or surface area) will increase by (1.5*1.5 – 1)*100 % = 125%
Note: Similarly A^{3 }(or volume) will increase by (1.5*1.5*1.5 – 1)*100 % = 237.5%
Answer: (A)
Where ever the word “melting” is used in mensuration, it means only one thing – equate the volume
The volume of the rectangular block = l*b*h = 21*77*24 cm^{3}
Now this volume will be equal to the volume of the sphere formed after melting the block
Volume of sphere = (4/3) * π * r^{3} = 21*77*24
Hence, r = 21 cm
Answer: (A)
The water rises by 5.6 cm. Take this 5.6 cm as the height of the cylindrical beaker and find its volume.
Volume of a cylinder = π*r*r*h = π * (7/2) * (7/2) * 5.6
Volume of the marbles (spherical in shape) = (4/3) * π * r^{3} = (4/3) * π * 0.7 * 0.7 * 0.7
No. of marbles dropped = Volume of beaker/Volume of a marble = 150
Answer: (B)
Let the radius of the big sphere be R.
Volume of a cone = (1/3) * π * R^{3} (since radius and volume are same as the radius of the sphere)
Let the radius of the smaller sphere = r
Then volume of cone = volume of smaller sphere
(1/3) * π * R^{3} = (4/3) * π * r^{3}
r : R = 1 : 2^{2/3}
Surface area of smaller sphere(s) = 4 * π * r^{2}
Surface area of larger sphere(S) = 4 * π * R^{2}
S/s = (r/R)^{2} = 1 : 2^{4/3}
Answer: (D)
When a cone is hollowed out from a cylinder, we get the above figure
The whole surface area of the remaining solid = Area of A + Area of B + Area of C
A = curved surface area of the cone
B = curved surface area of the cylinder
C = area of the cylindrical base
A = π * r * l, where l = slant height of the cone, which is or
Hence A = π*3*5 = 15π
B = 2πrh = 2π*3*4 = 24π
C = πr^{2} = π*3^{2} = 9π
The whole surface area of the remaining solid = 15π + 24π + 9π = 48π
Answer: (C)
Given, AB = 3 cm, BC = 6 cm and OF = 1 cm
Height of the cone (AC) = √(6^2 – 3^2 ) = 3√3 cm
Triangles ABC and CFO are similar (RHS similarity)
So, OC/BC = OF/AB
OC = 2 cm, therefore CG = 3 cm (OG = 1 cm)
Now, ABC and CEG are similar
GE/AB = CG/AC
So, GE =
Required volume = Volume of cone (CDE) – Volume of Sphere
= 3π – (4/3)π
= (5/3)π
Answer: (C)
If you have any doubt in this article, please drop a comment.
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