Question

Item 6

Question 1
You notice a hot air balloon descending. The elevation h (in feet) of the balloon is modeled by the function h(x)=-11x+330, where x is the time (in seconds) since you first noticed the hot air balloon.
Graph the function and specify its domain and range. Then interpret the slope and intercepts of the graph.

Answer 1

Domain

`x\in [0\,s,30\,s]`

Range

`h \in [0\,ft,330\,ft]`

The h-Intercept represents the initial height of the balloon. The balloon has an initial height of 330 feet.

The x-Intercept represents the time when the balloon reaches the ground. The balloon takes 30 seconds to descend and reach the ground.

The slope represents the ratio of the change in the elevation of the balloon to the change in time. A slope of -11 means that balloon descends 11 feet per second.

Explanation:

Both the time (domain), measured in seconds, and the height (range), measured in feet, are positive variables. The hot air balloon start at a height above and descends to the ground. Which means that domain and range of the function are, respectively:

Now we find the upper bound of the height of the hot air balloon. If we know that
`t = 0\,s`, then the upper bound is:

`h(0) = -11\cdot (0)+330`

`h(0) = 330\,ft`

Range

`h \in [0\,ft,330\,ft]`

Now we find the upper bound of time. If we know that
`h = 0\,ft`. then the upper bound is:

`-11\cdot x +330 = 0`

`11\cdot x = 330`

`x = 30\,s`

Domain

`x\in [0\,s,30\,s]`

The h-Intercept represents the initial height of the balloon. The balloon has an initial height of 330 feet.

The x-Intercept represents the time when the balloon reaches the ground. The balloon takes 30 seconds to descend and reach the ground.

The slope represents the ratio of the change in the elevation of the balloon to the change in time. A slope of -11 means that balloon descends 11 feet per second.