A meteorologist models the height of a hot air balloon launched from the ground. The model assumes the balloon travels vertically upwards and travels 450 metres in the first minute. Due to the decrease in temperature as the balloon rises, the balloon will continually slow down. The model suggests that each minute the balloon will travel only 82% of the distance travelled in the previous minute.

M1: Recognition that this is a geometric sequence with first term $a=450$ and common ratio $r=0.82$

M1: Correct use of sum formula $S_{10}=\frac{a(1-r^n)}{1-r}$ with correct substitution $S_{10}=\frac{450(1-0.82^{10})}{1-0.82}$

A1: $2160$ metres (or $2156.37\ldots$)

M1: Correct use of sum to infinity formula $S_{\infty}=\frac{a}{1-r}=\frac{450}{1-0.82}=2500$

A1: Conclusion that the balloon will not reach the required height as $2500 < 2520$ metres

A1: Any ONE of: • Horizontal motion is not taken into account • Rate of cooling will likely not be linear in reality • Balloon is considered a point mass / size of balloon not considered • Effects of wind/weather unlikely to be consistent • A discrete model has been used, whereas a continuous one may offer greater accuracy • Any other reasonable limitation

Note: Accept any other sensible answer that relates to real-world factors not accounted for in the model.