Cut that flat watermelon

We can use cauchy's inequality to find the maximum value

$(a_{1}^{x}+a_{2}^{x}+.......+a_{n}^{x})(b_{1}^{x}+b_{2}^{x}+.......+b_{n}^{x})\geq(a_{1}b_{1}+a_{2}b_{2}+......a_{n}b_{n})^{x}$ $(x^{2}+\frac{y^{2}}{4})(8^{2}+15^{2})\geq(8x+\frac{15y}{2})$

we get 17 as the final answer

put x=cos t............y=2sin t.....maxm value of 8cos t+15sin t=17

we can take x=cosa , y=2sina. we need to maximise 8cosa+15sina . its maximum is 17.

Set up Lagrange Multiplier such that 8x+7.5y is maximized subject to the constraint of the first equation. After finding the two gradients and equating the first to the second times some eigenvalue, the problem becomes a simple algebra problem. Find the value of x and y and plug them into the second equation to get a final value of 17.