Take the circumference of the earth at the equator to be 24,000 miles. An airplane taking off at the equator and flying west at 1000 miles an hour would land at exactly the same time that it started (why?). Moreover, the sun would not move in the plane’s sky during the flight. (Work this out visually in your mind.) Now, at what degree of latitude could a plane flying 500 miles per hour keep up with the sun in this way?
Set 08
Gertrude Goldfish
Two young men in tuxedos are walking down the street at about 10 pm. One of them is carrying a round goldfish bowl filled with water. There is a goldfish, named Gertrude, in the bowl. As they pass a round pool in the park, the goldfish gets very excited and jumps out of the bowl into the pool, right at the edge. She starts swimming due north. She hits the wall after she has swum exactly 30 feet. She heads east, and, after going 40 feet, she hits the wall again. Once she regains consciousness after this second collision, she calculates the diameter of the pool. What is it? And what is the story about the two guys in their tuxes?
Fences in Circular Region
In a circular field, place three fences to make four regions. The fences are all equal in length and their endpoints are on the circular boundary of the field. The four resulting regions have equal area, and the fences don�t intersect within the field.
Billy the Goat
Billy the goat is tied to the corner of Patty�s barn. The barn is 20 x 40 feet, and the rope is 50 feet long. No trees or other obstructions are in the way. What is the available area of grass that Billy can eat? (Draw a good picture of this area first.)
Algebra with Angles
If twice ∠A is subtracted from the supplement of ∠A, then the remaining angle exceeds the complement of ∠A by 4°. Find the size of ∠A.
Geometric and Arithmetric Sequences
There are two positive numbers that may be inserted between 3 and 9 such that the first three numbers are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is:
- 13½
- 11¼
- 10½
- 109½
Note: There are two other numbers that work, but they’re not both positive. If you go about this problem in a suitably erudite fashion, you’ll turn up this alternative solution too.
Sit-Ups
To make the team, you are going to have to do 89 sit-ups for the coach a week from today. You decide to work up to it. You will start by doing 3 sit-ups today (no sense rushing into things) and end on the 8th day with 89. You don’t know how many you will do tomorrow, but you decide that from the 3rd day on, the number of sit-ups you do will be the sum of what you did on the two preceding days. That is, the number you do on Wednesday will be the sum of the number you did on Monday and the number you did on Tuesday; the number you do on Thursday will be the sum of what you did on Tuesday and Wednesday, and so on. Find out how many sit-ups you should do tomorrow to make this work, so that you come out with 89 a week from today.
Hard Functional Equation, f(n) = n
The function f satisfies the functional equation
f(x) + f(y) = f(x + y) – xy – 1
for every pair x, y of real numbers. If f(1) = 1, then the number of positive integers n for which f(n) = n is:
- 0
- 1
- 2
- 3
- infinite
Defined Operations 2; Do Composition
If 3 h = 10, 7 h = 50, 5 h = 26; and 4 b = 1, 7 b = 2.5, 20 b = 9, then what is n
if n hb = 17.5?
Garden Crops
A large organic nursery has somewhere between 3 and 6 (inclusive) garden plots of herbs when it closes for the season in the fall. In each plot there are between 20 and 30 (inclusive) rosemary plants. If, typically, 10% of those plants don’t winter over successfully until spring, what would be the largest number of plants that could be lost during the winter?