Given that a solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8 : 27 : 27. We have to find the percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube.
Let us take the volume of the 5 cubes to be
Volume of the original cube =
Volume of the original cube =
Sides of the original cube =
Similarly, sides of the 5 smaller cubes = x , x , 2x , 3x , 3x .
Surface Area of a cube =
Surface Area of the original cube = (4x) × (4x) =
Surface area of the smaller cubes =
Sum of the surface areas of the smaller cubes =
Change in surface area =
%change =
Hence the percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to 50%.
A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically. The height of the cylinder is 3 cm, while its volume is Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is: (TITA)
Given that the height of the cylinder is 3 cm, while its volume is
A ball of diameter 4 cm is kept on top of a hollow cylinder standing vertically.
Since OPQ is the right angled triangle
We can find the OP = 1 cm.
We have to find the vertical distance of the topmost point of the ball from the base of the cylinder.
Since OP = 1, to reach the topmost point still it has to go 2 cm from the point O.
The vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is 2 + 1 + 3 = 6 cm
Let ABC be a right-angled triangle with BC as the hypotenuse. Lengths of AB and AC are 15 km and 20 km, respectively. The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is: (TITA)
Given that ABC be a right-angled triangle with BC as the hypotenuse.
Lengths of AB and AC are 15 km and 20 km, respectively.
We have to find the minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour.
We should first find the minimum distance in order to find the minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour.
Therefore minimum distance AD has to be found and then it should be divided by the 30 km per hour.
Using the idea of similar triangles
Area of the triangle ABC
Hence 12 kms is travelled at 30km per hour
The minimum possible time, in minutes, required to reach the hypotenuse from A at a speed of 30 km per hour is × 60 = 24 minutes
Key thing to be noted here is using Pythagoras theorem to find the altitude AD and then using Speed, Time and Distance formula to find the time.
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When rotated along hypo
Vol = πr²h/23 = π4.8²10/32 = 38.4π
When rotated along side = 6
Vol = π8²6/6 = 64π
When rotated along side = 8
Vol = π6²8/6 = 48π
So answer = 38.4"
Then the vertical distance, in cm, of the topmost point of the ball from the base of the cylinder is: (TITA)
× 60 = 24 minutes