, 2019

In 10 minutes, there are $10\times 60=600$ seconds.

If the machine is set to make one cut every 8 seconds, then $\text{600 s 8 s =75}$ pieces of rope will be cut off in 10 minutes.
Rope is fed into the machine at a constant rate of 2 metres per second.

If the machine is set to make one cut every 3 seconds, then the length of each piece of rope that is cut off is $2\times3=6$ metres.
Solution 1

Rope is fed into the machine at a constant rate of 2 metres per second, which is equivalent to $2\times60=120$ metres per minute.

If each piece of rope that is cut off is 30 m long, then the machine is set to make $\text{120 m 30 m =4}$ cuts per minute.

Solution 2

Rope is fed into the machine at a constant rate of 2 metres per second.

To cut off a piece of rope 30 m long, the machine must be set to make a cut every $\dfrac{30}{2}=15$ seconds.

If the machine makes a cut every 15 seconds, then it makes $\text{60 s 15 s =4}$ cuts per minute.

Solution 3

In part (b), the machine was set to make one cut every 3 seconds, and so the length of each piece of rope that is cut off is 6 metres.

Since rope is fed into the machine at a constant rate, to cut off a piece of rope that is times longer (30 m is 5 times longer than 6 m), the machine must be set to make each cut 5 times more slowly, or every $5\times 3=15$ seconds.

If the machine makes a cut every 15 seconds, then it makes $\text{60 s 15 s =4}$ cuts per minute.
Solution 1

If the machine is set to make 16 cuts per minute, or 16 cuts every 60 seconds, it will make one cut every $\text{60 s 16 =3.75}$ seconds. Rope is fed into the machine at a constant rate of metres per second.

Since the machine is set to make one cut every 3.75 seconds, then the length of each piece of rope that is cut off is $2\times3.75=7.5$ metres.

Solution 2

If the machine is set to make 16 cuts per minute, then every 60 seconds it is set to make 16 cuts.

The rope is fed into the machine at a constant rate of 2 metres per second, and so after 60 seconds, $60\times2=120$ metres of rope have passed through the machine.

During these 60 seconds, the 120 metres of rope is cut 16 times, and so the length of each piece of rope that is cut off is $\text{120 m 16 =7.5}$ metres.