1. **Problem statement:** We are given two circles where the difference of their circumferences is $4\pi$ and the difference of their areas is $24\pi$. We need to find the radii $r_{\text{groß}}$ (big circle) and $r_{\text{klein}}$ (small circle). 2. **Write down the formulas:** - Circumference of a circle: $U = 2\pi r$ - Area of a circle: $A = \pi r^2$ 3. **Set up the equations from the problem:** $$ U_{\text{groß}} - U_{\text{klein}} = 4\pi \implies 2\pi r_{\text{groß}} - 2\pi r_{\text{klein}} = 4\pi $$ $$ A_{\text{groß}} - A_{\text{klein}} = 24\pi \implies \pi r_{\text{groß}}^2 - \pi r_{\text{klein}}^2 = 24\pi $$ 4. **Simplify both equations by dividing by $\pi$:** $$ 2 r_{\text{groß}} - 2 r_{\text{klein}} = 4 \quad \Rightarrow \quad \cancel{2} r_{\text{groß}} - \cancel{2} r_{\text{klein}} = \cancel{4} 2 \implies r_{\text{groß}} - r_{\text{klein}} = 2 $$ $$ r_{\text{groß}}^2 - r_{\text{klein}}^2 = 24 $$ 5. **Use the difference of squares factorization:** $$ (r_{\text{groß}} - r_{\text{klein}})(r_{\text{groß}} + r_{\text{klein}}) = 24 $$ 6. **Substitute $r_{\text{groß}} - r_{\text{klein}} = 2$ from step 4:** $$ 2 (r_{\text{groß}} + r_{\text{klein}}) = 24 \implies r_{\text{groß}} + r_{\text{klein}} = 12 $$ 7. **Solve the system:** $$ \begin{cases} r_{\text{groß}} - r_{\text{klein}} = 2 \\ r_{\text{groß}} + r_{\text{klein}} = 12 \end{cases} $$ Add the two equations: $$ 2 r_{\text{groß}} = 14 \implies r_{\text{groß}} = 7 $$ 8. **Find $r_{\text{klein}}$ by substituting back:** $$ 7 - r_{\text{klein}} = 2 \implies r_{\text{klein}} = 5 $$ **Final answer:** - Radius of the big circle: $r_{\text{groß}} = 7$ - Radius of the small circle: $r_{\text{klein}} = 5$