A solid metallic cube is melted to form five solid cubes whose volumes are in the ratio 1 : 1 : 8: 27: 27. The percentage by which the sum of the surface areas of these five cubes exceeds the surface area of the original cube is nearest to10506020

To solve this problem, we need to find the surface area of each cube and compare it to the surface area of the original cube.

Let's assume the volume of the original cube is V. Since the volumes of the five cubes are in the ratio 1:1:8:27:27, we can assign the volumes as follows:

Volume of first cube = V/1 Volume of second cube = V/1 Volume of third cube = V/8 Volume of fourth cube = V/27 Volume of fifth cube = V/27

Now, let's find the side length of each cube. Since the volume of a cube is equal to the side length cubed, we can find the side length of each cube by taking the cube root of its volume.

Side length of first cube = (V/1)^(1/3) = V^(1/3) Side length of second cube = (V/1)^(1/3) = V^(1/3) Side length of third cube = (V/8)^(1/3) = (V/2)^(1/3) Side length of fourth cube = (V/27)^(1/3) = (V/3)^(1/3) Side length of fifth cube = (V/27)^(1/3) = (V/3)^(1/3)

Now, let's find the surface area of each cube. The surface area of a cube is given by 6 times the side length squared.

Surface area of first cube = 6(V^(1/3))^2 = 6V^(2/3) Surface area of second cube = 6(V^(1/3))^2 = 6V^(2/3) Surface area of third cube = 6((V/2)^(1/3))^2 = 6(V/2)^(2/3) Surface area of fourth cube = 6((V/3)^(1/3))^2 = 6(V/3)^(2/3) Surface area of fifth cube = 6((V/3)^(1/3))^2 = 6(V/3)^(2/3)

Now, let's calculate the sum of the surface areas of the five cubes:

Sum of surface areas = 6V^(2/3) + 6V^(2/3) + 6(V/2)^(2/3) + 6(V/3)^(2/3) + 6(V/3)^(2/3)

To find the percentage by which the sum of the surface areas exceeds the surface area of the original cube, we can use the following formula:

Percentage increase = ((Sum of surface areas - Surface area of original cube) / Surface area of original cube) * 100

Percentage increase = ((6V^(2/3) + 6V^(2/3) + 6(V/2)^(2/3) + 6(V/3)^(2/3) + 6(V/3)^(2/3) - 6V) / 6V) * 100

Simplifying the expression, we get:

Percentage increase = ((6V^(2/3) + 6V^(2/3) + 6(V/2)^(2/3) + 6(V/3)^(2/3) + 6(V/3)^(2/3) - 6V) / 6V) * 100

Now, we can substitute the given values and calculate the percentage increase.