Q. 6

# Find the total volume of 15 cubes whose edges are 16 cm, 17 cm, 18 cm, …., 30 cm respectively

Given that there are 15 cubes,

We see that their sides 16cm, 17cm..,30cm are in series.

To find the total volume, we know that the volume of a cube is simply l3 where l is the length of the side.

Volume of first cube = 16cm × 16cm × 16cm = 163 cm3

Volume of second cube = 17cm × 17cm × 17cm = 173 cm3

And so on.

We observe that this is in a series.

So S = 163 + 173 + 183 + … + 303

To find S,

Let S1 be 13 + 23 + …. + 303 with n = 30

Let S2 be 13 + 23 + …. + 153 with n = 15

S = S1-S2

To find S1

13 + 23 + …. + 303 with n = 30

Formula to find the sum of first n cubes of natural numbers is

S =  S1 = = S1 = 216225

To find the sum S2

13 + 23 + …. + 153 with n = 15

Formula to find the sum of first n cubes of natural numbers is

S =  S2 = = S2 = 14400

S = S1-S2

S = 216225-14400

S = 201825 cm3 The required sum S = 163 + 173 + 183 + … + 303 = 201825 cm3

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