Ratio between the areas of a circle and an inscribed rectangle

Text

A rectangle is inscribed in a circle; its base is $\dfrac{15}{8}$ of its height and its perimeter is $46$ cm. Find the ratio between the area of the circle and the area of the rectangle.

Solution

Let $b$ be the base and $h$ the height, with $b=\dfrac{15}{8}h$. From the perimeter: $2(b+h)=46\ \Rightarrow\ b+h=23\ \Rightarrow\ \frac{15}{8}h+h=\frac{23}{8}h=23\ \Rightarrow\ h=8,\ b=15.$ The diagonal (diameter) is $d=\sqrt{15^2+8^2}=\sqrt{289}=17$, so $r=8.5$. $\frac{\mathcal A_\text{circle}}{\mathcal A_\text{rect}}=\frac{\pi(8.5)^2}{15\cdot8}=\frac{72.25\,\pi}{120}=\frac{289\pi}{480}\approx1.89.$ $\boxed{\dfrac{\mathcal A_\text{circle}}{\mathcal A_\text{rect}}=\dfrac{289\pi}{480}\approx1.89}$