Carnival knowledge

Problem 159 — Being right on the money depends on your point of view, as this month’s problem by Imack Collins of Nashville, Ark., demonstrates.

“Stee-yep right up and tee-rye your luck,” barked Fibber McYack. The annual Tooterville Fruitfly Festival was under way, and McYack was in charge of the slingshot game. For $1, contestants received five chances to hit a stationary target, which would then drop a ball onto a moving conveyor belt that carried buckets. The target is 86.3 ft above the slingshot height and 80 ft above the lip of the bucket when the bucket is below the ball. If a ball lands in a bucket, the contestant wins a prize. McYack had arranged it so that, when a shot is fired, the nearest bucket would be at least 150 ft away from the point at which the target is directly overhead.

The contestant stands 250 ft in front of the target and releases the slingshot at an angle of 35 deg from the horizontal. The slingshot pellets travel as unresisted projectiles. Neglect weight, spin, and air resistance, and assume the ball is released the instant a pellet hits the target. If the buckets travel at 30 fps, and the pellet’s initial velocity is 130 fps, how far must the nearest bucket be for a ball to fall into it? Will Fibber McYack be giving out prizes or lollipops?

Technical consultant: Jack Couillard, Milwaukee.

Solution to last month’s problem 158

— You’re right on the money if you answered $262.50. Here’s the carryover:

From the problem we know the following:

1. Each guest has no money left after leaving the third and last room.

2. It costs $100 to enter a room and $100 to leave it.

3. Each guest’s money doubles after paying the entry fee in each room.

From these we can work backwards. In the last room, the following happens:

$0
+100 (Exit fee)
100 (Money after doubling)
÷ 2
50 (Money after paying entry fee)
+100 (Entry fee)
$150 (Money after leaving 2nd room)

Therefore, each guest had $150 upon entering the last room. Work backwards again for the second room:

$150
+100 (Exit fee)
250 (Money after doubling)
÷2
125 (Money after paying entry fee)
+ 100 (Entry fee)
$225 (Money after leaving first room)

Each guest had $225 upon entering the second room. Again, use the same method to see how much each guest had upon entering the first room:

$225
+100 (Exit fee)
325 (Money after doubling)
÷2
162.50 (After paying entry fee)
+100 (Entry fee)
262.50 (Amount to enter rooms)

Each guest started out with $262.50. Snodd’s charity exhibition may or may not have been a financial success, depending on out-of-court settlements!

Contest winner — Congratulations to Steven Orth of West Buxton, Maine, who won our January contest by having his name drawn from the 153 correct responses out of a total of 164 for that month. A TI-68 calculator is in the mail to him.

The TI-68 Advanced Scientific Calculator by Texas Instruments can solve five simultaneous equations with real and complex coefficients and has 40 number functions that can be used in both the rectangular and polar coordinate systems. Other functions include formula programming, integration, and polynomial root finding. The calculator also features a last-equation replay function that lets you double-check your work.

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