π in the sky

Problem 192 — Getting to the root of a problem can clear up areas of doubt, as this month’s problem by Steve Zhang of Aloe, Ore., demonstrates.

Deep in the Pentagon, a highly classified meeting between the joint chiefs of staff and their new radar supplier was in progress.

“I’m pleased to announce we’ve found a radar net system that’s both cost-effective and fully functional,” stated Brig. Gen. Brass. “Without further ado, here’s Mortimer Blynd of Tri-Gen/Git-Rite Inc.”

“The new Failsafe Radar Integrated Tracking System (FRITS) consists of four towers that each have a sweep radius of 10√3 miles ,” stated Blynd. “These towers (A, B, C, D) form a square, (see diagram), with an area of 900 square miles. Together, *ahem*, with some negligible overlap and one very small blind spot, *ahem*, they can monitor this area.”

Given the measurements stated, what is the area of the blind spot? Can the enemy hide a tank or a munitions plant in it?

Technical consultant, Jack Couillard, Menasha, Wis.

Solution to last month’s problem 191 — Your ETA is right on target if you answered Lottie and her boyfriend get wet. Here’s the watered-down analysis:

First, let’s find out if the water balloon will strike the Ferris wheel or pass over it.

Let:
y1 = Maximum height attained by the water balloon, ft
t = Time from launch to maximum height, min
x = Horizontal distance from launch point to the point directly beneath the balloon’s zenith, ft
v0 = Initial velocity of water balloon, given as 60 fpm
Φ = Water balloon launch angle, given as 60 deg
g = Acceleration due to gravity, 32.2 ft/sec2

From physics, we know that the time it takes a projectile to reach its zenith is:

The balloon’s maximum elevation is:

x = v0(cosΦ)t = 48.4 ft (1)

The horizontal distance the water balloon has traveled at this time is:

Since the closest point of the Ferris wheel is 55 ft away, the balloon is descending by the time it reaches the unsuspecting couple. At this point you can draw a diagram to scale and note that the balloon strikes the wheel at a horizontal distance between 65 and 70 ft. Use (1) to find that the time is between 2.16 and 2.33 sec.

The Ferris wheel turns at 1 rpm, so between the time frame allotted, the wheel has turned 13 or 14 degrees in its rotation. Lottie and company are clearly the ones in the line of fire.

Contest winner — Congratulations to Joe Kohler of Cleveland, who won our January contest by having his name drawn from the 11 contestants who answered correctly out of a total of 32 entrants for that month. A TI-85 calculator is in the mail to him.

The TI-85 Graphing Calculator from Texas Instruments solves for any variable in an equation, can solve 30 simultaneous equations, and finds the roots of a polynomial up to the 30th order. It handles complex numbers in addition to matrices, vectors, lists, and strings. You can perform graphic investigations of almost any type of problem — functions as well as parametric, polar, and differential equations.

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Fun with Fundamentals: Problem 191