Question

A hot air balloon starts at an elevation of 300 feet. Then, it ascends at a rate of 600 feet per minute.
Remember, the slope is the change in vertical distance for a given change in horizontal distance. We can use any two points on the line to find the slope. Label the coordinates of the two points on the graph below.

x
y
(
,
)(
,
)
0
1
2
3
4
5
6
7
8
9
10
0
300
600
900
1,200
1,500
1,800
2,100
2,400
2,700
3,000
Number of minutes
Elevation in feet
Slope
=
vertical distance
horizontal distance
=
?

Answer 1

To find the slope of the hot air balloon's ascent, we can use two points that represent the elevation at different times.

The starting point (at time \( t = 0 \) minutes) is:
- Point 1: \( (0, 300) \)

To find another point, we can calculate the elevation after a specific number of minutes. Since the balloon ascends at a rate of 600 feet per minute, we can find the elevation after, for example, 5 minutes:
- Elevation after 5 minutes = Initial elevation + (rate of ascent × time)
- Elevation after 5 minutes = \( 300 + (600 \times 5) = 300 + 3000 = 3300 \)

So, the second point will be:
- Point 2: \( (5, 3300) \)

Now we can label the two points:
- Point 1: \( (0, 300) \)
- Point 2: \( (5, 3300) \)

Next, we can calculate the slope using the formula:

\[
\text{Slope} = \frac{\text{change in vertical distance}}{\text{change in horizontal distance}} = \frac{\Delta y}{\Delta x}
\]

Here:
- \( \Delta y = 3300 - 300 = 3000 \) feet (the change in elevation)
- \( \Delta x = 5 - 0 = 5 \) minutes (the change in time)

Plugging these values into the slope formula gives:
\[
\text{Slope} = \frac{3000}{5} = 600
\]

Thus, the slope of the hot air balloon's ascent is 600 feet per minute.

In summary:
- Point 1: \( (0, 300) \)
- Point 2: \( (5, 3300) \)
- Slope: \( 600 \) feet per minute.