André trains for the marathon by running laps around a circular track with a $100 \mathrm{~m}$ radius. To cover $42 \mathrm{~km}$, the number of laps André needs to run is between:
(a) 1 and 10 ;
(b) 10 and 50 ;
(c) 50 and 100 ;
(d) 100 and 500 ;
(e) 500 and 1000 .
The correct option is (c).

The circumference of a circle with radius $r$ is $2 \pi r$. Thus, in each lap, André covers $2 \pi \times 100 \text{ m} = 200 \pi \text{ m}$. Therefore, the number of laps André needs to complete $42 \text{ km} = 42,000 \text{ m}$ is

$$
\frac{42000}{200 \pi} = \frac{210}{\pi}
$$

Now we can conclude the problem in two ways.

1a) Since $3 < \pi < 4$, we get $\frac{1}{4} < \frac{1}{\pi} < \frac{1}{3}$, therefore, multiplying everything by 210, we have

$$
52.5 = \frac{210}{4} < \frac{210}{\pi} < \frac{210}{3} = 70
$$

and we conclude that André must complete between 52 and 70 laps to cover the $42 \text{ km}$.

2b) The approximation of $\pi$ to the second decimal place is 3.14. Thus,

$$
\frac{210}{\pi} \approx \frac{210}{3.14} \approx 66.88
$$

and we conclude that André must complete between 66 and 67 laps to cover the $42 \text{ km}$.