Varsity Math, Week 120

Varsity Math 120

________________

With the new year approaching, the passage of time is on people’s minds.

________________

Polar

Breakfast, Lunch, or Dinner?

Just as team member Malcolm is finishing a meal, he notices the minute hand is on a minute mark exactly one minute ahead of the hour hand.

Which meal is Malcolm finishing?

The 2018 Puzzle

The coach has an interesting problem to celebrate the coming year.

“You have 2018 distinct points in space. Prove that there is a plane that divides the points into two sets, with 1009 points on each side of the plane.”

momathlogohorizontalWSJLogo

Solutions to week 119

In Solar Cap Versus Polar Cap, the solar cap and polar cap have equal areas. The pole in How Tall Is the Pole is ≈ 180.75 feet tall.

Solar Cap Versus Polar Cap answer explained:Polar
Let the compass radius be r. On a sphere of radius R the area of a spherical cap of half-angle θ is 2π(1 – cosθ)R2 (θ is shown in the figure.) Also from the figure, cosθ = 1 – 2sin2(θ/2) = 1 – 2r2/(2R)2. Putting these together gives the area of the spherical cap as πr2, independent of R.

How Tall Is the Pole answer explained:Polar
Let h be the height of the pole. r = 20996000 feet is the earth’s radius of curvature, and d is the extra length of chain needed to accommodate the pole. Then, from the figure,
    d = 2r(tanθ – θ) = 2r(θ3/3 + 2θ5/15 + …) ≈ 2rθ3/3
    h = r(secθ – 1) ≈ rθ2/2 = (9d2r/32)1/3 ≈ 180.749… feet.

Another way to solve the problem is to use elementary trigonometry and the trigonometric equation solver found here: http://www.wolframalpha.com/widgets/view.jsp?id=27b8624af4f632acc36d65c4a7eb715.

Recent Weeks

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

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

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

Come back next week for answers and more puzzles.