Question

A Hot air balloon can hold 90,000 cubic feet of air. It is being inflated at a rate of 6,000 cubic feet per minute . The total cubic feet of air a (t) is a function of the time in minutes t .
Identify the independent and dependent variables .
What values of the domain and range make sense for this situtaion ?
Write a function to represent the total amount of air. Then determine the total amount of air in 6 minutes .

Answer 1

$\bf \stackrel{\textit{cubic feet of air}}{\stackrel{\textit{dependent}}{a(t)}}\qquad \stackrel{\stackrel{\textit{time in minutes}}{independent}}{t}~\hspace{7em}a(t)=6000t$

the "t" variable is independent, namely it can change values "freely", and a(t) is dependent, because it depends on those values on "t" to get its own value.

if we make a table of values for minutes, we know the balloon is inflating at 6000 ft³ each passing minute.

1 minute......................... 6000(1) ft³

2 minutes......................6000(2) ft³

3 minutes......................6000(3) ft³

4 minutes......................6000(4) ft³

t minutes......................6000(t) ft³

before getting in the domain and range, let's find out how much air after 6 minutes.

a(6) = 6000(6)

a(6) = 36000 ft³.

now, what domain, namely values for "t" make sense?

well, we know the balloon can only hold up to 90000 ft³, and on every passing minute is filling up by 6000 ft³, after how many minutes will it be filled up?

90000/6000 = 15, after 15 minutes.

so at 0 minutes, t = 0, the balloon is empty, and 15 minutes later, t = 15, the balloon is filled up, [ 0 , 15 ].

what values for the range, namely a(t), makes sense?

well, we know when the balloon is empty is holding 0 ft³, and when is full it has 90000 ft³, [0 , 90000].