Pitched diameter

Problem 219 -- Sinking your teeth into a situation doesn't always get you up the hill, as this month's problem by Wesley Richardson of Lexington, Ky., demonstrates.

The expensively clad moderator Festus Fopp strode onto the stage as the live audience of the highly rated game show, "Let's Make a Fool," got underway.

"As all of you know, we're in the final round for $50,000," stated Fopp. "This week we have a very special task for our contestants. You all see this ramp. It's at a 20° angle with the horizontal. Now the cash is in a bag hanging at the top. The contestants will ride a two-wheeled bike up the ramp and try to grab the bag with a hook. The specially- made bicycle that will be used in today's program was furnished by the well-known Wauble Bicycle Co."

The Wauble XXX bicycle has the following specifications:
• Weight: 10 kg.
• 43 teeth on pedal sprocket and 32 teeth on rear-wheel sprocket.
• Foot pedal is 15 cm from the crank hub center.
• Tires on both wheels have 68-cm diameters.
• Assume the bike starts from rest on the ramp, the pedal crank can be repositioned to any angle, no drag or frictional losses, and that the contestant can only use his weight. (He cannot grab the handle bars and push down with his legs.)

If the first contestant's weight is 67 kg, can he pedal up the ramp? What is the steepest angle that he could start the bike moving uphill?

Solution to last month's problem 218 -- You know all the angles if you answered 87.5 ft. Here's the way the laser bounces.

Let:
a = Distance of laser from ceiling, given as 100 ft
b = Distance of bowl from ceiling, given as 60 ft
c = Horizontal distance of mirror from laser, ft
l = Distance of bowl from laser, given as 140 ft

We can make the following proportion:

The party was a success until the mirror fell down and smashed on the bar.

CONTEST WINNER – Congratulations to Mike Pederson of Worthington, Minn., who won our July contest by having his name drawn from the 206 contestants who answered correctly out of a total of 231 entrants for that month. A Maple V Release 5 mathematics software package is in the mail to him.

Maple V Release 5 is an interactive computer algebra system that provides a mathematical environment for manipulating symbolic algebraic expressions, arbitrary-precision numerics, graphics, and programming. Its library features over 2,700 functions that are used in many scientific and engineering applications. Features include a MATLAB link, which lets users execute MATLAB commands from inside the software; a spreadsheet function in which Maple V operations can be performed; and an HTML export function to create web pages. The software is available in Windows 95, Windows NT, Windows 3.11, Macintosh, Power Macintosh, plus UNIX and Linux platforms.

This month's problem submitter receives an EL-546L Direct Algebraic Logic calculator from Sharp Electronics. The EL-546L has an extra-large, 10-digit, two-line LCD display that lets you see the full equation on one line and the answer to the equation on the other. An algebra and playback function enables you to enter an equation and substitute numeric values or correct errors. Another feature is the full-term equation entry function, which enables the calculator to display a fraction equation exactly as it appears in a textbook. The EL-546L has dual power and can automatically switch from solar to battery power in low light.

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Fun with Fundamentals: Problem 218