## Task 4 - 160814

Peter presents his friend Fritz with the following problem:

"Given a circle whose diameter is equal to the Earth's diameter, and a second concentric circle whose circumference is $1 \mathrm{~m}$ longer than the circumference of the first circle. Determine the distance between the two circle lines!"

After a brief moment of thought, Fritz names this distance and claims:

"If the first circle has only the diameter of a pinhead ( $1 \mathrm{~mm}$ ), and the circumference of the second concentric circle is again $1 \mathrm{~m}$ longer than that of the first circle, then the distance between these two circles is exactly the same as in your problem."

Is Fritz's claim correct?

How large is the distance between the concentric circle lines in both cases?
If $d$ is the Earth's diameter and $u$ (in $\mathrm{m}$) is the circumference of the given smaller circle, then $\pi \cdot d = u$.

If $x$ is the distance sought in Peter's problem, then the diameter of the second (larger) circle is $d + 2x$. On the other hand, its circumference is $u + 1 = \pi d + 1$; thus, $\pi(d + 2x) = \pi d + 1$, which leads to $2 \pi x = 1$, and $x = \frac{1}{2 \pi}$.

Similar conclusions apply to Fritz's problem if $d (=1 \mathrm{~mm}) = 0.001 \mathrm{~m}$, because the obtained value $x = \frac{1}{2 \pi}$ does not depend on $d$. Therefore, Fritz's claim is correct. The sought distance in both cases is $\frac{1}{2 \pi}$ (in $\mathrm{m}$), which is approximately $16 \mathrm{~cm}$.

Solutions of the 1st Round 1976 adopted from [5]

### 5.18.2 2nd Round 1976, Class 8