42) Swathi has 15 ice cubes different sizes 9cm, 10 cm, 11cm......... 23 cm. How much volume of ice cubes can be used to prepare some fruit juice with these ice cubes?
Answer:
The total volume is the sum of the volumes of all cubes. The volume of a cube with side 'a' is \(a^3\). We need to find the sum:
\[ S = 9^3 + 10^3 + 11^3 + \dots + 23^3 \]
We can write this sum as the difference of two sums of cubes starting from 1:
\[ S = (1^3 + 2^3 + \dots + 23^3) - (1^3 + 2^3 + \dots + 8^3) \]
The formula for the sum of the cubes of the first n natural numbers is \(\sum_{k=1}^{n} k^3 = \left[ \frac{n(n+1)}{2} \right]^2\).
Part 1: Sum up to 23.
\[ (1^3 + \dots + 23^3) = \left[ \frac{23(23+1)}{2} \right]^2 = \left[ \frac{23 \times 24}{2} \right]^2 = (23 \times 12)^2 = (276)^2 = 76176 \]
Part 2: Sum up to 8.
\[ (1^3 + \dots + 8^3) = \left[ \frac{8(8+1)}{2} \right]^2 = \left[ \frac{8 \times 9}{2} \right]^2 = (4 \times 9)^2 = (36)^2 = 1296 \]
Final Calculation:
\[ S = 76176 - 1296 = 74880 \]
The total volume of the ice cubes is 74,880 cm³.