Graphics Reference
In-Depth Information
33 If
a
and
b
arepositiv, provethat
(
ab
)
(
a
b
).
ab
a
b
Under what circumstances can there be equality?
34 For any positivenumbr
a
, provethat 2
a
(1/
a
).
35 'In a remote village, the (only) greengrocer uses a scale-balance and
a 1 kg weight. Unfortunately, as a result of an accident, the balance
breaks and the greengrocer, in repairing it, does not get the point
of balance exactly in the middle, but slightly offset.
A customer enters and asks for 2 kg of apples. The greengrocer
places his 1 kg weight in the left-hand pan and fills the right-hand
pan with apples until the scales balance. He then empties the apples
into a brown paper bag. Now he puts the 1 kg weight to the
right-hand pan and fills up the left-hand pan with apples until the
scales balance. He then adds these apples to those in the brown
paper bag and gives the lot to the customer.' (Law of moments:
when the scale pans balance, the weight in one pan times the length
of its scale arm is equal to the weight in the other pan times the
length of its scale arm.) Decide between
(1) The customer gets 2 kg of apples.
(2) The customer gets
more
than 2 kg of apples.
(3) The customer gets
less
than 2 kg of apples.
36 For a given perimeter, what shape of rectangle gives the greatest
area?
37 If
x
and
a
areboth positivenumbrs, what arethearithmtic and
geometric means of the numbers
x
and
a
/
x
?
If
x
0 and
x
is defined by
x
(
x
a
/
x
), show that
a
x
x
, for
n
2.
38 If
a
and
b
are given positive numbers with
a
b
, and two
sequences of positive numbers are defined by
