To the left is a circle
with an inscribed square. Obviously, there isn’t room for another
nonoverlapping square of the same size within the circle. But suppose that you
divided the square into n2
smaller squares, each with side length 1/n.
Would one of those smaller squares fit in the space between the large square
and the circle? As shown to the left, this works if n = 16 and the large square were divided into 256 smaller
squares. But it would work for smaller values of n, too.
What is the smallest value
of n such that one of the smaller
squares would fit between the larger square and the circle?
In the chart, color each square according to the clues below.
The triangle at left lies on a flat surface and is pushed at the top vertex. The
length of the congruent sides does not change, but the angle between the two
congruent sides will increase, and the base will stretch. Initially, the area
of the triangle will increase, but eventually the area will decrease,
continuing until the triangle collapses.
is the maximum area achieved during this process? And, what is the length of
the base when this process is used to create a different triangle whose area is
the same as the triangle above?
Look at the panel of
elevator buttons shown. Can you find a set of three buttons whose centers
form the vertices of a right triangle and whose numbers are the side lengths of
a right triangle? (The classic 3-4-5 right triangle doesn’t work, because the
3, 4, and 5 buttons don’t form a right triangle on the elevator panel.)
And after you’ve found one
set, can you find another?
do I love thee? Let me graph the ways!
you come up with one or more equations to graph a heart on the coordinate
plane? The equations can be rectangular, polar, or parametric.
Can you shift your heart so the graph or its interior includes the point (2, 14)?
Equations to solve in your
Is this a joke? Not if you can
multiply the first equation by 6,751 and the second by 3,249 in your head, and
not if you use a second, simpler method.
The Fibonacci sequence is shown below, with each term equal to the
sum of the previous two terms. If you take the ratios of successive
terms, you get 1, 2,
, and so on. But as you proceed through the sequence, these ratios get
closer and closer to a fixed number, known as the Golden Ratio.
1, 1, 2, 3, 5, 8, 13, …
Using the rule that defines the Fibonacci sequence, can you determine the value of the Golden Ratio?
A plywood sheet is 45 by 45
inches. What is the approximate diameter of the log the sheet was made from?
The diameter d of a circle equals ,
where C is the circumference, but
please do not make a mistake. The diameter of the log is not .