Let A 1 = x , y : x ≤ y 2 , x + 2 y ≤ 8 and A 2 = x , y : x + y ≤ k . If 27 (Area A 1 ) =…

$A_1=\left\{\left(x,y\right):\left|x\right|\le y^2,\left|x\right|+2y\le 8\right\}$ and

$A_2=\left\{\left(x,y\right):\left|x\right|+\left|y\right|\le k\right\}$

Area $\left(A_1\right)=2\left[{\int }_0^2{y}^{2}dy+{\int }_2^4\left(8-2y\right)dy\right]$

$=2\left[{\left(\frac{y^3}{3}\right)}_0^2+{\left(8y-y^2\right)}_2^4\right]$

Area $\left(A_1\right)=2\times \frac{20}{3}=\frac{40}{3}$

Area $\left(A_2\right)=4\times \frac{1}{2}{k}^2$

Area $\left(A_2\right)=2k^2$

Given $27\left(\mathrm{Area} A_1\right)=5\left(\mathrm{Area} A_2\right)$

$\Rightarrow 27\times \frac{40}{3}=5\times 2k^2$

$\Rightarrow k=6$