5.2 Developing models 219
He will only consider blocks that are cuboid
in shape and have dimensions that are
whole numbers of metres.
a Consider a block with dimensions
2 m × 2 m × 6 m. Calculate the minimum
possible time that it would take to
roll the block through 360°. [1]
b Consider a block with dimensions
1 m × 4 m × 6 m. Calculate all the
different possible distances that the
block could travel in one 360° revolution,
according to the different initial
orientations. [1]
c If a 24 m^3 block is to travel at least
610 m, what is the smallest possible
number of 90° turns that will be needed?610 m
[3]
d What dimensions for a 24 m^3 block
will allow for the smallest possible
time to move it 610 m? State the time
it will take, to the nearest minute. [4]
He decides that he needs 61 m^3 of stone
for the next season. He can only move
a maximum of 24 m^3 at a time. It takes
5 minutes for him to return the 610 m
distance without a stone.
e Show that it is possible for him to move
exactly 61 m^3 of stone in less
than 500 minutes. [4]
f He realises that by transporting more
than 61 m^3 of stone in total, he can reduce
the overall amount of time. However, he
does not want to move any more than
70 m^3 or there will be too much waste.What set of block sizes should he move
to minimise his total time? [2]Cambridge International A & AS Level Thinking
Skills 9694/31 Paper 3 Q3 May/June 20113 (Harder task) A motor race consists
of 60 laps of 5 km each. Some of the
specifications for the Marlin team car are as
follows:
fuel consumption: 1 litre/km
fuel tank capacity: 160 litres
refuel rate: 15 litres/second
pit stop time: 10 seconds plus time to refuel
average speed (no fuel): 75 seconds/lap
speed with fuel: 0.12 seconds slower/lap
for each 5 litres of fuel carried
It may be seen that the car cannot carry
enough fuel to complete the race without a
pit stop. However, the car goes more slowly
the more fuel it carries. The fuel gauge is
very accurate, so it can effectively be run
down to zero before refuelling. (Hint: in
order to calculate the average lap time for
each section you may use the average fuel
load. Assume the race is broken into equal
distances between pit stops.)
How many pit stops should the car make
to complete the race in the fastest possible
time – 1, 2 or 3?
4 A shop sells three types of nuts:Brazil nuts: 80¢/kg
walnuts: 70¢/kg
hazelnuts: 40¢/kg
The shopkeeper makes 50% profit on each
type of nut. She wishes to sell mixed nuts
at 60¢ per kg. What proportion should the
mix of the three nuts be if she is to make
50% profit on the mixed nuts? Is there one
answer or a range of answers? If so, which
contains the most even mix of nuts?
Can you generalise the result?
Answers and comments are on pages 327–30.