Varsity Math, Week 119

Varsity Math 119

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The coach shares two puzzles involving the North Pole with the team.

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Polar

Solar Cap Versus Polar Cap

Imagine that you have a giant compass set to a radius of 1,000 miles, and use it to draw a circle on the surface of the sun. Without changing the opening of the compass, draw a second circle with its center on the North Pole.

Which is larger, the area on the sun’s surface enclosed by the first circle, or the area on the Earth’s surface enclosed by the second circle? Model both bodies as perfectly smooth spheres.

Pole

How Tall Is the Pole?

Imagine a large chain with length exactly equal to the Earth’s circumference. With one foot added to its length around the world, it just reaches over a pole placed at the North Pole, as shown in the figure. Because the Earth is slightly squashed at the poles, the important parameter is the radius of curvature at the North Pole, which is approximately 20,996,000 feet.

How tall is the pole to the nearest hundredth of a foot?

momathlogohorizontalWSJLogo

Solutions to week 118

In Good Relations, Alex is Chris’s son-in-law or daughter-in-law. In Square Purchases, there were five gifts costing as follows: $1 + $9 + $25 + $36 + $784 = $855.

Good Relations answer explained:
Chris’s mother-in-law’s son is either a husband or brother-in-law to Chris. Alex’s father-in-law cannot be Chris’s brother-in-law because Chris has no nieces or nephews. Therefore, Chris’s mother-in-law’s son is Chris’s husband. That husband is Alex’s father-in-law, making Alex either Chris’s son-in-law or daughter-in-law.

Square Purchases answer explained:
There were five gifts costing as follows: $1 + $9 + $25 + $36 + $784 = $855. To use the 7, the smallest squares without repeating a digit are 576, 729, and 784. With 576, the smallest square using 2 and not 5, 6, or 7 is 289, which gives a larger sum than 855. With 729, there is no way to get the digit 5 in a small enough square without using the 2.

Recent Weeks

Week 118: Relations Puzzle & Square Purchases, solutions to Fifty-Fifty & Random Toss

Week 117: Fifty-Fifty & Random Toss, solutions to Cutting the Domino & Divide by Three

Week 116: Cutting the Domino & Divide by Three, solutions to Thanksgiving Split & Easy as Pie

Week 115: Thanksgiving Split & Easy as Pie, solutions to Primes and Products & Two Pints of Cider

Week 114: Primes and Products & Two Pints of Cider, solutions to Prime Presents 1 & Prime Presents 2

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

Come back next week for answers and more puzzles.