## ________________

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### Circle Stack

Two cylinders 80 centimeters long — one with a radius of 17 centimeters, the other with a radius of 9 centimeters — are packed tightly into the shallowest possible box that is 50 centimeters wide and 80 centimeters long.

*How deep is the box?*

### Three Circles, One Distance

In the diagram, circle *c* has radius 3, and *A* and *B* are any distinct points on *c* with less than 90 degrees of arc between them. Circle *a* has center *A* and goes through *B*, while *b* has center *B* and goes through *A*. Point *E* is the intersection of *a* and *b* inside *c*, and *F* is the other intersection point of *a* and *c*. Point *D* is the other intersection of line *FE* with *c*.

*What is the distance from D to E? If you’re stuck, Coach Newton’s advice is to remember that when a problem leaves something arbitrary, choose it to make things as easy as possible. Can you find a good angle for arc AB? Maybe one that makes DE and EF equal?*

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## Solutions to week 56

**Equitable Area**. Notice that region *K* is symmetric about the diagonal of the square. Therefore, in order for triangle *ADF* to have area equal to region *K*, triangle *ACF* must have half the area of triangle *ADF*. But *ADF* and *ACF* have the same height *AD*, so therefore *CF* must be **1/2** of *FD*.

**Angular Equity**. In order for all of the angles at vertex *A* to be equal, angle *DAF* must be 30°. If we take side *AD* to be one, then *FD* is the tangent of 30°, or √3/3. That leaves (3-√3)/3 for *CF*. And as in the previous solution, the area of *ADF* is to the area of *K* as *FD* is to *twice* *CF*, so the area of *ADF* divided by the area of *K* is √3/2(3-√3) = **(√3+1)/4**.

## Recent Weeks

**Week 56**: Equitable Area & Angular Equity, solutions to Diagonalization & Regionalization

**Week 55**: Diagonalization & Regionalization, solutions to Watch for Falling Nuts & Fall Out of Line

**Week 54**: Watch for Falling Nuts & Fall Out of Line, solutions to Hexadiagonal & Hexintersection

**Week 53**: Hexadiagonal & Hexintersection, solutions to Mean Triangle & As Easy as 4132

**Week 52**: Mean Triangle & As Easy as 4132, solutions to Rhombarium & Well Trained

Links to all of the puzzles and solutions are on the Complete Varsity Math page.

**Come back next week** for answers and more puzzles.

[asciimathsf]