Question 2

The volume of gas, $V$ liters, in a high-altitude research balloon is recorded as it rises through the atmosphere. The altitude is measured as $h$ km above the launch site.

When the balloon is exactly 5 km above the launch site, the volume is measured as 30 liters.

When the balloon is exactly 10 km above the launch site, the volume is measured as 50 liters.

Given that the relationship between the volume and altitude is modelled by the equation

V^2 = p + qh^2

where $p$ and $q$ are constants,

find the exact value of $p$ and the exact value of $q$ .

According to the model, the volume of the balloon reaches 80 liters at an altitude of exactly $H$ km.

Making your method clear,

find the value of $H$ , giving your answer to one decimal place.

(Solutions relying entirely on calculator technology are not acceptable.)