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03 Dec 2021, 10:00
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A gold ingot in the shape of a cylinder is melted and the resulting molten metal is molded into few identical conical ingots. If the height of each conical ingot is half the height of the original cylinder and the area of the circular base of the cone is one fifth that of the cylinder, how many conical ingots can be made?
A. 60
B. 10
C. 30
D. 20
E. 40
Are You Up For the Challenge: 700 Level Questions
A. 60
B. 10
C. 30
D. 20
E. 40
Are You Up For the Challenge: 700 Level Questions
03 Dec 2021, 10:36
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The volume of the prism-cylinder is found by multiplying the Base Area by the Perpendicular - Straight Line length (Height) in which the base is extended.
In the case of a Cylinder, we find the Area of the Circular Base and multiply it by the perpendicular Height in which the Circular Base is extended.
Let the Radius of the Cylinder = 5 and the Perpendicular Height by which the base is extended be 10
(Since (pi) will be used to find the volume in both instances for the Cylinder and Cones, whatever answer we would have found using (pi) is the same if we don’t use (pi))
Total Volume of the Cylinder reduced by a Factor of (pi) =
(r)^2 (h) = (5)^2 (10) = 250
In the case of the cone-pyramid, the circular edge of the circular base will extend and converge to meet at a pointed Apex
Because of this fact, the volume of a cone is (1/3)rd the volume of a cylinder with the same radius and perpendicular height ——>
Volume of Cone = (1/3) (Area of Base) (perpendicular height from the Center of the circular base to the Apex)
The height of each individual cone is half of the height of the cylinder (1/2) * (10) —— h = 5
And the Area of the Cone’s circular base is (1/5)th that of the Cylinder
Cylinder’s Circular Base Area —- (r)^2 = (5)^2 = 25
Thus, Each cone’s Circular Base Area = (1/5) (25) = 5
Therefore, the volume of each individual cone will be:
(1/3) (5) (5) = 25/3
The number of cones that can be created from the melted cylinder is given by:
(Volume of Cylinder) / (Volume of each individual Cone) = (250) / (25/3) = (750 / 25) = 30
Answer: 30 cones
C
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In the case of a Cylinder, we find the Area of the Circular Base and multiply it by the perpendicular Height in which the Circular Base is extended.
Let the Radius of the Cylinder = 5 and the Perpendicular Height by which the base is extended be 10
(Since (pi) will be used to find the volume in both instances for the Cylinder and Cones, whatever answer we would have found using (pi) is the same if we don’t use (pi))
Total Volume of the Cylinder reduced by a Factor of (pi) =
(r)^2 (h) = (5)^2 (10) = 250
In the case of the cone-pyramid, the circular edge of the circular base will extend and converge to meet at a pointed Apex
Because of this fact, the volume of a cone is (1/3)rd the volume of a cylinder with the same radius and perpendicular height ——>
Volume of Cone = (1/3) (Area of Base) (perpendicular height from the Center of the circular base to the Apex)
The height of each individual cone is half of the height of the cylinder (1/2) * (10) —— h = 5
And the Area of the Cone’s circular base is (1/5)th that of the Cylinder
Cylinder’s Circular Base Area —- (r)^2 = (5)^2 = 25
Thus, Each cone’s Circular Base Area = (1/5) (25) = 5
Therefore, the volume of each individual cone will be:
(1/3) (5) (5) = 25/3
The number of cones that can be created from the melted cylinder is given by:
(Volume of Cylinder) / (Volume of each individual Cone) = (250) / (25/3) = (750 / 25) = 30
Answer: 30 cones
C
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03 Dec 2021, 11:38
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Radius of cylinder = R
Radius of cone = r
Height of cylinder = H
Height of cone = h
Given
pi*R²=1/5*pi*r²
H = 1/2*h
Cylinder is melted into x number of cones. This implies that volume of cylinder is equal to volume of x number of cones.
pi*R²H = x*1/3*pi*r²*h
1/5*pi*r²*(1/2*h) = x*1/3*pi*r²*h
Cancelling the like terms
x=30
Posted from my mobile device
Radius of cone = r
Height of cylinder = H
Height of cone = h
Given
pi*R²=1/5*pi*r²
H = 1/2*h
Cylinder is melted into x number of cones. This implies that volume of cylinder is equal to volume of x number of cones.
pi*R²H = x*1/3*pi*r²*h
1/5*pi*r²*(1/2*h) = x*1/3*pi*r²*h
Cancelling the like terms
x=30
Posted from my mobile device
