Varsity Math, Week 124

Varsity Math 124

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This week, the coach has two challenges to hone the team’s geometric and visualization skills.

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Cutting Triangles

Cutting the Triangle

Cut an equilateral triangle into three pieces of identical shape where:

(a) They are all the same size.
(b) They are all of different sizes.
(c) Two are the same size and the third is a different size.

Pieces may be turned over.

Square Repair

A square has had a 45°-30°-105° triangle cut from it and pasted to its right edge, as seen in the figure below.

Cut the figure into two pieces in a different way that will still allow assembly into the original square.

Square Repair

momathlogohorizontalWSJLogo

Solutions to week 123

In Mining Mishap, the miner was 66 inches up the ladder. In Pulley Puzzle, the 99-weight will stay at rest upon release. The 45-weight accelerates upward at g/10 and the 55-weight accelerates downward at g/10, where g is the acceleration due to gravity.

Mining Mishap answer explained:
If the rock falls from a distance d above the top of the ladder, it takes a time of √(2d/g) seconds to reach the top of the ladder. If the miner is h feet up the ladder and climbs at 4 feet per second then he would take (10 – h)/4 seconds to reach the top of the ladder. Thus, 2d/g = (10 – h)2/16. The rock takes √[(2d + 20)/g] seconds to reach the bottom of the ladder and the miner would take h/4 seconds to reach the bottom of the ladder. Thus, (2d + 20)/g = h2/16. If we take the difference of the two equations, we get 20/g = 10(2h – 10)/16, which leads to h = 5.5 feet. Thus, the miner was 66 inches up the ladder.

Pulley Puzzle answer explained:
If the 99-weight remains stationary then the tension in the rope around the larger wheel must be 99g and the tension in the rope around the smaller wheel must be T = 99g/2, where g is the acceleration due to gravity. The upward acceleration of the 45-weight must equal the downward acceleration of the 55-weight and, indeed, a45 = T/45 – g = 11g/10 – g = g/10 and a55 = g – T/55 = g – 9g/10 = g/10.

Pulley Puzzle Solution

Recent Weeks

Week 123: Mining Mishap & Pulley Puzzle, solutions to Minimum Racket Wear & Rolling Circles

Week 122: Minimum Racket Wear & Rolling Circles, solutions to Tennis Exhibition & Losing Streak

Week 121: Tennis Exhibition & Losing Streak, solutions to Breakfast, Lunch, or Dinner? & The 2018 Puzzle

Week 120: Breakfast, Lunch, or Dinner? & The 2018 Puzzle, solutions to Solar Cap Versus Polar Cap & How Tall is the Pole

Week 119: Solar Cap Versus Polar Cap & How Tall is the Pole, solutions to Relations Puzzle & Square Purchases

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

Come back next week for answers and more puzzles.