If x and y are two positive real numbers such that 8x3 +27y3= 730 and 2x2y+3xy2= 15,then evaluate 2x+3y
Given,
8x3 +27y3= 730
⇒ (2x)3 + (3y)3 = 730
⇒ (2x + 3y)(4x2 + 9y2 - 6xy) = 730
⇒ (2x + 3y)(4x2 + 9y2 + 12xy - 18xy) = 730
⇒ (2x + 3y)[(2x + 3y)2 - 18xy] = 730
⇒ (2x + 3y)3 - 18xy(2x + 3y) = 730 ... (1)
Again, we have 2x2y+3xy2= 15
⇒ xy (2x + 3y) = 15
On putting the above value in (1), we get
⇒ (2x + 3y)3 - 18 × 15 = 730
⇒ (2x + 3y)3 = 730 + 270
⇒ (2x + 3y)3 = 1000
⇒ (2x + 3y) = 10