During a hot-­‐air balloon festival, data were gathered on the height y of a balloon after x minutes.

Question

Time (minutes)	Height (meters)
0	14
2.2	80
3.4	116
4	134
4.6	152

C. (__/ 3) 

If the balloon continues to rise at the same rate, use function notation to express the height after 12 minutes. 

D. (__/ 3) 

Provide the domain given that the maximum allowed height is 500 meters.

Answer 1

It looks like the table might represent a linear relationship, which is sort of implied by the phrase "continues to rise at the same rate".

Let's use rows of the table to calculate some slopes.

Using the 1st two rows: change in y is 80 - 14 = 66, change in x = 2.2 - 0 = 2.2.

So slope is 66/2.2 = 30.

Let's do one more to sort of confirm our suspicions.

Use rows 2 and 4.

Change in y (actually change in height) is 134 - 80 = 54, change in time is 4 - 2.2 = 1.8.

Slope is: 54/1.8 = 30.

Since slopes are the same this is probably a linear relationship which means:

m = 30 and y-intercept is (0,14).

So in terms of x and y:

y = 3x + 14.

Writing as a function using h and t:

h(t) = 30t + 14.

Now you can find h(0), h(2.2), h(3.4), ... to confirm linear.

h(12) = 30(12) + 14 = 374

If the maximum height is 500 what value of t generates 500?

500 = 30t + 14

30t = 486

t = 16.2

That means the values of t that make sense for this function are 0 less than or equal to t which is less than or equal to 16.2. That is the domain of the function.

Here is a Desmos graph confirming the above. Take a look by selecting this URL, right-clicking, and choosing Go-to from the drop-down menu:

desmos.com/calculator/x5ieyusgde