Varsity Math, Week 140

Varsity Math 140

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The coach provides these two reasoning problems to prepare the math team.

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Nick’s Birthday

A while back, Nick stated: “Sometime during last year, I was still 21; in two days I’ll be in my 25th year.”

What day of the year is Nick’s birthday and on what day of the year is he speaking?Fuse

Burning Ropes

You have two ropes and some matches. The ropes burn irregularly like fuses when lit at either end. The first rope burns in e = 2.71828…hours and the second rope burns in √2 = 1.414213… hours.

Produce a time interval as close as possible to 1 hour.

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Solutions to week 139

In Pondering Productivity, the hens in both farms are equally productive. Smith’s farm produces 245 eggs in 6 days; Jones’s farm produces 6,125 eggs in 24 days. In Cornfield Planning, cut out quarter circles on the corners with radius = 100/(2 + √π) = 26.5079… feet. This gives an area-to-perimeter ratio of 100/(2 + √π) = 26.5079…

Pondering Productivity answer explained:
(a) Eight of Smith’s hens will lay 7 × (7/6) eggs in 6/5 days for a rate of (7/8) × (7/6) ÷ (6/5) = 245/288 eggs per hen per day. Six of Jones’s hens will lay 5 × (7/6) eggs in 8/7 days for a rate of (5/6) × (7/6) ÷ (8/7) = 245/288 eggs per hen per day. The hens from both farms are equally productive.
(b) With 48 hens Smith gets 245/6 eggs daily so must wait 6 days to get 245 eggs.
(c) With 300 hens Jones gets 6125/24 eggs daily so must wait 24 days to get 6125 eggs.

Cornfield Planning answer explained: Cornfield
Cut out quarter circles on the corners as shown in the figure. The area of such a shape is A = 10000 – 4r2 + πr2. The perimeter of the shape is P = 400 – 8r + 2πr. The ratio A/P takes on a maximum value of 100/(2+√π) = 26.5079… when r = 100/(2+√π) = 26.5079… If the cornfield were either a circle or a square, the ratio A/P would be 25.

Recent Weeks

Week 139: Pondering Productivity & Cornfield Planning, solutions to Close to a Quart & Math Party

Week 138: Close to a Quart & Math Party, solutions to Room for One More & End Points

Week 137: Room for One More & End Points, solutions to Sevens and Elevens & Three-Digit Squares

Week 136: Sevens and Elevens & Three-Digit Squares, solutions to Swimming Workout & Batting Order

Week 135: Swimming Workout & Batting Order, solutions to Maximization & Multiplication Table

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

Come back next week for answers and more puzzles.