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27 Jan 2021, 09:45
Kudos
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
65%
(hard)
Question Stats:
63% (03:01) correct
37%
(02:51)
wrong
based on 43
sessions
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A solid metal cylinder of height equal to double the base diameter is melted and from this molten mass three identical spheres are made. What is the percentage change in the total surface area? (Area of a sphere \(= \frac{4}{3}\pi r^3\). The surface area of a sphere \(=4\pi r^2\))
(A) 70%
(B) 50%
(C) 26%
(D) 20%
(E) 15%
(A) 70%
(B) 50%
(C) 26%
(D) 20%
(E) 15%
27 Jan 2021, 10:29
Kudos
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Let radius of cylinder = R
and, radius of each sphere = r
Height of cylinder, H = 2*diameter of cylinder = 2*2R = 4R
Volume of cylinder = \(\pi*R^2H\) = \(\pi*R^2*4R\) = \(4\pi*R^3\)
Total surface area of cylinder = \(2\pi*R^2+2\pi*RH\) = \(10\pi*R^2\)
Since three spheres are made out of the cylinder,
3*Vol. (each sphere) = Vol. (cylinder)
\(3*\frac{4}{3}\pi*r^3\) = \(4\pi*R^3\)
Simplifying, we get, r = R
Surface area of each sphere = \(4\pi*r^2\) = \(4\pi*R^2\)
Surface area of all three spheres = \(3*4\pi*R^2\) = \(12\pi*R^2\)
Percentage difference in surface area = \(\frac{Area(All Spheres)-Area(Cylinder)}{Area(Cylinder)}*100\)% = \(\frac{12\pi*R^2-10\pi*R^2}{10\pi*R^2}*100\)% = \(\frac{2}{10}*100\)% = 20%
Ans is D IMO
and, radius of each sphere = r
Height of cylinder, H = 2*diameter of cylinder = 2*2R = 4R
Volume of cylinder = \(\pi*R^2H\) = \(\pi*R^2*4R\) = \(4\pi*R^3\)
Total surface area of cylinder = \(2\pi*R^2+2\pi*RH\) = \(10\pi*R^2\)
Since three spheres are made out of the cylinder,
3*Vol. (each sphere) = Vol. (cylinder)
\(3*\frac{4}{3}\pi*r^3\) = \(4\pi*R^3\)
Simplifying, we get, r = R
Surface area of each sphere = \(4\pi*r^2\) = \(4\pi*R^2\)
Surface area of all three spheres = \(3*4\pi*R^2\) = \(12\pi*R^2\)
Percentage difference in surface area = \(\frac{Area(All Spheres)-Area(Cylinder)}{Area(Cylinder)}*100\)% = \(\frac{12\pi*R^2-10\pi*R^2}{10\pi*R^2}*100\)% = \(\frac{2}{10}*100\)% = 20%
Ans is D IMO
03 May 2021, 10:07
Kudos
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Step 1: find the volume of the cylinder and break it down into 3 spheres (there is a misprint: the Volume of 1 sphere =
(4/3) (pi) (r)^2 )
Cylinder has radius = r
Cylinder has height = (2) * (Diameter) = (2) (2r) = 4r
Volume of Cylinder = (pi) (r)^2 (H)
V = (pi) (r)^2 (4r)
V = (pi) (4) (r)^3
Now break this volume, melt it down, and create 3 spheres
Each sphere will have Volume = (1/3) * (pi) * (4) * (r)^3
Which is general formula for the Volume or the Sphere given radius = r
The Surface Area of a sphere is = (4) * (Area of 4 circles with radius r ) = (4) (pi) (r)^2
We have 3 of these Spheres, so multiplying this result by 3 to add up all 3 surface areas of the spheres:
(12) (pi) (r)^3
(2nd) calculate the surface area of the original right cylinder
The Surface Area of a Right cylinder = (2 Areas of Identical Base and Top Circles) + (Circumference of one Base Circle) * (Height of Cylinder)
SA = (2) (pi) (r)^2 + (2) (pi) (r) * (4r)
SA = (10) (pi) (r)^2
The NEW surface area of the 3 spheres is = 12 * (pi) (r)^2
An Increase from Original 10 to New 12 corresponds to a 20% increase in the Surface area
Answer: 20% increase
Posted from my mobile device
(4/3) (pi) (r)^2 )
Cylinder has radius = r
Cylinder has height = (2) * (Diameter) = (2) (2r) = 4r
Volume of Cylinder = (pi) (r)^2 (H)
V = (pi) (r)^2 (4r)
V = (pi) (4) (r)^3
Now break this volume, melt it down, and create 3 spheres
Each sphere will have Volume = (1/3) * (pi) * (4) * (r)^3
Which is general formula for the Volume or the Sphere given radius = r
The Surface Area of a sphere is = (4) * (Area of 4 circles with radius r ) = (4) (pi) (r)^2
We have 3 of these Spheres, so multiplying this result by 3 to add up all 3 surface areas of the spheres:
(12) (pi) (r)^3
(2nd) calculate the surface area of the original right cylinder
The Surface Area of a Right cylinder = (2 Areas of Identical Base and Top Circles) + (Circumference of one Base Circle) * (Height of Cylinder)
SA = (2) (pi) (r)^2 + (2) (pi) (r) * (4r)
SA = (10) (pi) (r)^2
The NEW surface area of the 3 spheres is = 12 * (pi) (r)^2
An Increase from Original 10 to New 12 corresponds to a 20% increase in the Surface area
Answer: 20% increase
Posted from my mobile device
