Question

leah and her friends go on a hot air balloon ride. The graph below shows the rate at which the balloon rises. What is the constant of proportionality?

Answer 1

Answer: 125
Reasoning: to get 4 to 500 you need to divide 500 by 4 and you should be able to get the answer.

Answer 2

1. Rise (change in elevation) = 1000 feet

2. Run (change in time) = 8 minutes

3. Slope (constant of proportionality) = Rise / Run = 1000 feet / 8 minutes = 125 feet per minute

The constant of proportionality in the context of the graph provided, which shows a hot air balloon's elevation over time, is the rate at which the balloon rises per unit of time. This is also known as the slope of the line on the graph.

To find the constant of proportionality, we follow these steps:

1. Identify two clear points on the line. In the context of a graph, these points are typically where the line crosses the gridlines.

2. Use the formula for slope (which is also the constant of proportionality here) given by:

`\[ \text{slope} = \frac{\text{change in elevation (rise)}}{\text{change in time (run)}} \]`

3. Calculate the 'rise' by finding the difference in elevation (y-values) between the two points.

4. Calculate the 'run' by finding the difference in time (x-values) between the two points.

5. Divide the rise by the run to find the slope.

Let's assume the line passes through the origin (0,0) since it appears to start there and use another point on the graph where the line crosses the gridlines clearly, for instance, at (8 minutes, 1000 feet).

Now, applying these steps:

1. Point 1 (x1, y1)

= (0 minutes, 0 feet)

2. Point 2 (x2, y2)

= (8 minutes, 1000 feet)

3. Rise = y2 - y1

= 1000 feet - 0 feet

= 1000 feet

4. Run = x2 - x1

= 8 minutes - 0 minutes

= 8 minutes

5. Slope (constant of proportionality) = Rise / Run

= 1000 feet / 8 minutes

Calculating the slope will give us the constant of proportionality. Let's do that calculation.

The constant of proportionality for the hot air balloon's rate of ascent is 125 feet per minute. This means that for every minute that passes, the balloon rises 125 feet. Here's the calculation:

1. Rise (change in elevation) = 1000 feet

2. Run (change in time) = 8 minutes

3. Slope (constant of proportionality) = Rise / Run = 1000 feet / 8 minutes = 125 feet per minute