The sum of three numbers is 80 and the sum of the squares of these three numbers is 2,650. If the product of the three numbers is 13,500, then find the sum of the cubes of the given three numbers.
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Correct answer: D
Explanation
given :
a + b + c = 80
a² + b² + c² = 2650
abc = 13500
According to the formula:
a² + b² + c² = (a + b + c)² − 2(ab + bc + ca)
a³ + b³ + c³ − 3abc = (a + b + c)[(a + b + c)² − 3(ab + bc + ca)]
2650 = 80² − 2(ab + bc + ca)
2650 = 6400 − 2(ab + bc + ca)
2(ab + bc + ca) = 3750
ab + bc + ca = 1875
(a + b + c)² − 3(ab + bc + ca)
= 80² − 3 × 1875
= 6400 − 5625 = 775
a³ + b³ + c³ − 3abc
= 80 × 775 = 62000
3abc = 3 × 13500 = 40500
a³ + b³ + c³= 62000 + 40500= 102500