Answers to assignments 327
5.2 Developing models
1 Let us assume that Duane walks x km. It
doesn’t matter whether this is done as a
single stage or they swap bike and walk
several times – it is only important how
far in total each walks and rides.
Duane’s total journey time is
x
6+ ()12
15
−x.Mervin’s total journey time is
x
20+^12
4
−xIf they arrive at town at the same time:
x
6+ ()12
15
−x = x
20+ ()12
4
−xMultiplying both sides by 60:
10 x + 4(12 − x) = 3x + 15(12 − x)
or 10x + 48 − 4x = 3x + 180 − 15x
Simplifying:
18 x = 132 or x = 7.33 km
The total time is:
x
6+ ()12
15
−x = 1.22 + 0.31= 1.53 hours (1 hour 32 minutes)
We still have to convince ourselves that
arriving at the same time is the best
strategy. Suppose Duane (the faster
walker) walks the whole way. It takes
him two hours. Clearly any strategy in
which Duane walks more than 7.33 km
will result in a slower time (nearer to two
hours). It is even worse if Mervin walks
further as he is a slower walker.
2 a Two orientations are possible.
• Along a 2 × 2 face:
time = 1 + 1 + 1 + 1 = 4 mins
• Along a 6 × 2 face:
time = 3 +^13 + 3 +^13 = 6 mins 40 secs
So four minutes is the minimum possible
time.
b Three orientations are possible.
• Along the 1 × 4 face:
distance = 1 + 4 + 1 + 4 = 10 m15 minutes between villages. Thus it is at
village 1 (nearest Matsberg) from 8.00 to
8.05, at village 2 from 8.20 to 8.25 and
at village 3 from 8.40 to 8.45, arriving
at Aaland at 9.00. They both arrive at
village 2 at 8.20, so C is correct.
3 Dunrovia has 6 points – this can only
be obtained by two wins and one loss.
Similarly, Arbadia’s and Brindling’s score
of 4 points can only be achieved by one
win, one draw and one loss. Crittle’s
score of 2 points can only be obtained by
two draws and a loss.
a Crittle drew two matches. Two other
teams drew a match each; thus Crittle
must have drawn with both Arbadia
and Brindling. Two matches were
drawn.
b Crittle drew two matches and lost one;
Dunrovia won two and lost one. Thus
Dunrovia must have beaten Crittle.
c This leaves three games unaccounted
for: D vs A, D vs B and A vs B, none
of them draws. If Brindling beat
Dunrovia, Dunrovia must have
beaten Arbadia, and Arbadia must
have beaten Brindling.
4 If Chico’s bill was $3 more than the
average, Andy and Benita would have
paid $13.50 each ($12 + $^32 ) if all had
paid the average. Thus Chico’s individual
bill was $16.50.
5 If the border is the same all the way
around, and there is one more square
vertically than horizontally, the
difference between the two dimensions
must be the same as one square. Thus the
squares are 0.3 m × 0.3 m. The border is
0.1 m on each side.
5 The total expenses were 2 × $400 (two
weeks’ fixed costs) + $1400 for materials,
or $2200 in total. Thus the profit was
$2700, or $900 each. Bill had paid out
$800, so Fred owes him $1700. Harry had
paid out $1400, so Bill owes him $2300,
leaving $900 for himself.