## Task 4 - 090834

Let $K_{1}, K_{2}, K_{3}, K_{4}$ be four concentric circles, for whose radii $r_{1}, r_{2}, r_{3}$, and $r_{4}$

$$
r_{4}-r_{3}=r_{3}-r_{2}=r_{2}-r_{1}=r_{1} \quad \text { holds. }
$$

Determine the ratio of the area of $K_{1}$ to the areas of the three annuli formed by $K_{1}$ and $K_{2}$, $K_{2}$ and $K_{3}$, and $K_{3}$ and $K_{4}$!
From the equations for the radii, it follows that $r_{2}=2 r_{1}, r_{3}=3 r_{1}, r_{4}=4 r_{1}$.

The area $A_{1}$ of the inner circle $K_{1}$ is

$$
A_{1}=\pi r_{1}^{2}
$$

The area $A_{2}$ of the first circular ring is

$$
A_{2}=\pi\left[\left(2 r_{1}\right)^{2}-r_{1}^{2}\right]=3 \pi r_{1}^{2}
$$

The area $A_{3}$ of the second circular ring is

$$
A_{3}=\pi\left[\left(3 r_{1}\right)^{2}-\left(2 r_{1}\right)^{2}\right]=5 \pi r_{1}^{2}
$$

The area $A_{4}$ of the third circular ring is

$$
A_{4}=\pi\left[\left(4 r_{1}\right)^{2}-\left(3 r_{1}\right)^{2}\right]=7 \pi r_{1}^{2}
$$

It follows that:

$$
A_{1}: A_{2}: A_{3}: A_{4}=\pi r_{1}^{2}: 3 \pi r_{1}^{2}: 5 \pi r_{1}^{2}: 7 \pi r_{1}^{2}=1: 3: 5: 7
$$

The four areas are in the ratio $1: 3: 5: 7$.