Problem 505

Since $(p, q)$ lies on the line $y=\frac{2}{5} x$, then $q=\frac{2}{5} p$. The given rectangle has two sides on the axes, so has width $p$ and height $q$. Therefore, the area of the rectangle equals $p q=p \cdot \frac{2}{5} p=\frac{2}{5} p^{2}$. Since we are told that the area of the rectangle is 90, then $\frac{2}{5} p^{2}=90$ or $p^{2}=\frac{5}{2}(90)=225$. Since $p>0$, then $p=\sqrt{225}=15$.