A grasshopper jumps from its position at (x,y) over the point (a,b) to the point that is diametrically opposite to (x,y), i.e., the three points are collinear, and the grasshopper’s new position is just as far from (a,b) as it was before it jumped. What are the coordinates of this new point, in terms of a, b, x, and y?
Set 03
Spinning in Circles
We humans are traveling around in circles without even knowing it. We travel a circle around the axis of the earth once a day, and an almost-circle around the sun once a year. Considering each trip independently, how fast do we travel in each circle? (Assume we’re on the equator.) Here are some helpful data:
Give each answer in kilometers per hour.
Find Diameter of Circle
AB is a diameter of a circle. Tangents AD and BC are drawn so that segments AC and BD intersect at a point E on the circle. If AD = a and BC = b, with a ≠ b, then the diameter AB is:
| a. | |a – b| | ||
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| c. | √ab | ||
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Giant Clock, Bug Speed
The largest four-faced clock in the USA is the Allen Bradley Clock Tower in Milwaukee, Wisconsin. The diameter of its face is 40 feet 3.5 inches, and its minute hand is 20 feet in length. If you are a bug sitting on the very tip of the outer end of the minute hand, how fast are you traveling?
Stack of Water Pipes
Three of the new water pipes for Main Street are stacked as shown. The external diameter of each pipe is 12 inches. How tall is the stack (h in the picture)?
Circle Around 3 Squares
The radius of the smallest circle containing the symmetric figure composed of the three squares shown at the right is:
- √2
- √1.25
- 1.25
- 5√17
16 - none of these
Walking to Make Triangle
Nigel walks x miles due west, turns 150° to his left and walks 3 miles in the new direction. If he finishes at a point 2 miles from his starting point, then x is:
- 13/10
- 39/10
- 1.5√3
- not uniquely determined by the given information.
Function as Average
Let xk = (-1)k for any positive integer k. Let f(n) = (x1 + x2 + … + xn)/n, where n is a positive integer. Give the range of this function.
- 0
- 1/n (where n is any positive integer)
- 0 and -1/n (where n is any odd positive integer)
- 0 and 1/n (where n is any positive integer)
- 1 and 1/n (where n is any odd positive integer).
Okla Homer Baggage
When Homer Smith and his wife Okla flew to Rome, they had together 94 pounds of baggage. Homer paid $15.00 and Okla paid $20.00 for the excess weight of their baggage. If Homer had made the trip by himself with the combined baggage of both of them, he would have had to pay $135.00 for excess baggage. How many pounds of baggage can one person take along without charge?
Defined Operation; Find Formula
If 4 → 3, 10 → 6, 7 → 4.5, then what is the formula for x → ?