Question

if x and y are two positive real numbers such that 8x³+27y³=730and 2x²y+3xy²=15 then find 2x+2y=?

Answer 1

$8 {x}^{3} + 27 {y}^{3} = 730 \\ {(2x + 3y)}^{3} = 8 {x}^{3} + 27 {y}^{3} + 18xy(2x + 3y) \\ {(2x + 3y)}^{3} = 730 + 18xy(2x + 3y) \\ 2 {x}^{2} y + 3x {y}^{2} = 15 \\ xy(2x + 3y) = 15 \\ (2x + 3y) = \frac{15}{xy} \\ { (\frac{15}{xy}) }^{3} = 730 + 18xy \times \frac{15}{xy} \\ \: \: \: \: \: \: \: \: \: \: \: \: = 730 + 270 \\ \: \: \: \: \: \: \: \: \: \: \: \: = 1000 \\ \frac{15}{xy} = 10 \\ xy = \frac{3}{2} \\ (2x + 3y) = 15 \times 2 \div 3 = 10\\ 2x + 3y = 10$