Question

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

Answer 1

To find the total volume of $15$ cubes we have to apply the formula for sum of cubes of $n$ natural numbers and that is:

Sum of cubes of $n$ natural numbers is $S_n={\left[\frac{n(n+1)}{2}\right]}^2$

The series of total volume of $15$ cubes is as follows:

${16}^3+{17}^3+{18}^3+.......+{30}^3$

We have to find the sum of cubes from $1$ up to $30$ and subtract the sum of cubes of series from $1$ up to $15$ from that as shown below:

$V=(1^3+2^3+3^3+.......+{30}^3)-(1^3+2^3+3^3+.......+{15}^3)={\left[\frac{30(30+1)}{2}\right]}^2-{\left[\frac{15(15+1)}{2}\right]}^2$

$={\left[\frac{30\times 31}{2}\right]}^2-{\left[\frac{15\times 16}{2}\right]}^2=(15\times 31{)}^2-(15\times 8{)}^2=(465{)}^2-(120{)}^2=216225-14400=201825$

Hence, the total volume is $201825$.