## Task 3 - V10923

Inge draws 5 concentric circles, starting with the smallest one $\left(r_{1}=2 \mathrm{~cm}\right)$.

How should she choose the radii if the initial circle and the resulting circular rings are to have equal areas?
The radius of the smallest circle is $r_{1}=2 \mathrm{~cm}$. The radii of the remaining concentric circles, whose corresponding annular rings are to have the same area as the initial circle, are $r_{2}, r_{3}, r_{4}$, and $r_{5}$. It is required that

$$
F_{1}=\pi r_{1}^{2}=F_{2}=\pi\left(r_{2}^{2}-r_{1}^{2}\right)=F_{3}=\pi\left(r_{3}^{2}-r_{2}^{2}\right)=F_{4}=\pi\left(r_{4}^{2}-r_{3}^{2}\right)=F_{5}=\pi\left(r_{5}^{2}-r_{4}^{2}\right)
$$

This means $\pi r_{1}^{2}=\pi\left(r_{2}^{2}-r_{1}^{2}\right)$; dividing by $\pi$ then yields

$$
\begin{array}{r}
r_{1}^{2}=r_{2}^{2}-r_{1}^{2} \Rightarrow 2 r_{1}^{2}=r_{2}^{2} \Rightarrow r_{2}=r_{1} \sqrt{2} \\
r_{2}^{2}-r_{1}^{2}=r_{3}^{2}-r_{2}^{2} \Rightarrow 2 r_{2}^{2}-r_{1}^{2}=r_{3}^{2} \Rightarrow r_{3}=r_{1} \sqrt{3} \\
r_{3}^{2}-r_{2}^{2}=r_{4}^{2}-r_{3}^{2} \Rightarrow 2 r_{3}^{2}-r_{2}^{2}=r_{4}^{2} \Rightarrow r_{4}=r_{1} \sqrt{4} \\
r_{4}^{2}-r_{3}^{2}=r_{5}^{2}-r_{4}^{2} \Rightarrow 2 r_{4}^{2}-r_{3}^{2}=r_{5}^{2} \Rightarrow r_{5}=r_{1} \sqrt{5}
\end{array}
$$

Inge must choose the following radii for the construction: $r_{1}=2 \mathrm{~cm}, r_{2}=2 \sqrt{2} \approx 2.8 \mathrm{~cm}, r_{3}=2 \sqrt{3} \approx 3.5$ $\mathrm{cm}, r_{4}=2 \sqrt{4}=4 \mathrm{~cm}$, and $r_{5}=2 \sqrt{5} \approx 4.5 \mathrm{~cm}$.