Download Geometry Extensions Extensions of Algebra and Geometry

Extensions of Algebra and Geometry
1) Is it possible for a positive number to exceed its reciprocal by exactly 1? One number that comes close
is 8/5, because 8/5 – 5/8 is 39/40. Is there a fraction that comes closer?
2) Find the number that is two thirds of the way (a) from −7 to 17; (b) from m to n.
a c
 is not equivalent to
b d
ac
a
that
is in fact between the numbers and
bd
b
3) It is well known that
ac
. Suppose that a, b, c, and d are all positive. Show
bd
c
a c
, while  is not.
d
b d
4) Imagine covering an unlimited plane surface with a single layer of pennies, arranged so that each penny
touches six others tangentially. What percentage of the plane is covered?
5) The sides of a right triangle are x − y, x, and x + y, where x and y are positive numbers, and y < x. Find
the ratio of x to y.
M at hem at ics 1
6) What is the exact value of the expression x2 – 5 when x = 2+√5?
Sam’s Warehouse, a member pays $25 a year for membership, and buys at t he
t ore prices. A non-member does not pay t he membership fee, but does pay an
7) Using a driver on the 8th tee, which is on a plateau 10 yards above the level fairway, Dale hits another
al 5% above t he store prices. Under what condit ions would it make sense t o
buy
fine shot. Explain why the quadratic function y = 10.0 + 0.5x − 0.002x 2 describes this parabolic trajectory,
ership?
shown in the figure below. Why should you expect this tee shot to go more than 250 yards? Estimate the
length
calculator
to 1find
more
accurate
2
ch t he graphs
of yof
= this
x 2 +shot,
5, ythen
= xuse
− your
4, and
y = x2 +
onat he
same
axes. value.
WhatHow does this trajectory relate
to
the
trajectory
for
the
drive
on
the
previous
hole?
ect of the value of c in equat ions of t he form y = x 2 + c?
y
........................................................
math I I . Again using a driver on t he 8t h
....................
.............
.............
..........
.......•...
.........
.
.
.
.
.
.
.
.
........
h is on a plateau 10 yards above t he level ....................
.......
.......
......
......................
...... ..
.
.
.
.
.
.... ......
Dale hit s anot her fine shot . Explain why ..........
.... x
2
rat ic funct ion y = 10.0+ 0.5x − 0.002x describes t his parabolic trajectory, shown
findexpect
the length
theshot
shott without
a calculator,
must set y equal to 0 and solve for
gure above. (Continuation)
Why should To
you
this of
tee
o go more
t han 250you
yards?
2
why,
how to
arrive or
at xt o−find
250xa=more
5000.accurat e value.
e t he lengt hx.ofExplain
t his shot
, tand
henshow
use your
calculat
(a)
The
next
step
in
the
solution
process
is
to
add
to both hole?
sides of this equation.
s t his t raject ory relat e t o t he t rajectory for the drive on the previous
(b) Complete the solution by showing that the length of the shot is
. How does this
t inuat ion) To
find twhich
he lengt
h of 268.6,
t he shot
without
calculator,
must set y
number,
is about
compare
witha your
previousyou
calculation?
2
0 and solve(c)
forComment
x. Explain
why,
and
show
how
t
o
arrive
at
x
−
250x
=
5000.is its significance?
on the presence of the number 125 in the answer. What
2
next st ep in t he solut ion process is t o add 125 t o bot h sides of t his equat ion.
s t his number
chosen?
8) The
graph of a quadratic function intersects the x-axis at 0√and at 8. Draw two parabolas that fit this
plet e t he solut
ion
by showing
that the length
of tHow
he shot
125 + 20625.
How
description
and find equations
for them.
manyisexamples
are possible?
number, which is about 268.6, compare with your previous calculat ion?
ment on the9)presence
t he number
125
in t he
answer.axis
What
is it s significance?
There is aofunique
parabola
whose
symmetry
is parallel
to the y-axis, and that passes through the
three points (1, 1), (−2, −2), and (0, −4). Write an equation for it.
2
2
2
ph t he equat10)
ions
on tany
he same
syst emmust
of axes:
, y = 0.5x
, y pass
= 2xthrough
, and them? Explain.
Given
three points,
thereybe= a xparabola
that will
2
What is t he effect of a in equat ions of t he form y = ax ?
t he point s A = (4, 0), B = (4, 5), C = (0, 7), and D = (0, 0).
...
........... ..................
.........
...........
..
...........
.........
..
.................
...
. ...
.
......... ..
11) What number is exactly midway between 23 − √17 and 23 + √17? What number is exactly midway
b  b2  4ac
b  b2  4ac
between
and
?
2a
2a
12) Robin is mowing a rectangular field that measures 24 yards by 32 yards, by pushing the mower around
and around the outside of the plot. This creates a widening border that surrounds the un-mowed grass in
the center. During a brief rest, Robin wonders whether the job is half done yet. How wide is the uniform
mowed border when Robin is half done?
13) A stop sign — a regular octagon — can be formed from a 12-inch square sheet of metal by making
four straight cuts that snip off the corners. How long, to the nearest 0.01 inch, are the sides of the
resulting polygon?
16) One stick is twice as long as another. You break the longer stick at a random point. Now you have
three sticks. What is the probability that they form a triangle?
17) One stick is three times as long as another. You break the longer stick at a random point. Now you
have three sticks. What is the probability that they form a triangle?
18) Find the sum 1 + 2 + 3 + ··· + 100.
19) Find the sum of A1+ A3 + A2 + ··· + An
20) Without using a calculator, find the value of
when
and
.