Q. 6

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

Answer :

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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