A hot air balloon climbs into the air at a steady rate per minute. The graph illustrates the hot air balloon's ascent. Which example models the correct slope?

Question

(1 point)
Responses

m=10−620−12
m is equal to the fraction with numerator 10 minus 6 and denominator 20 minus 12

m=2−48−4
m is equal to the fraction with numerator 2 minus 4 and denominator 8 minus 4

m=12−00−24
m is equal to the fraction with numerator 12 minus 0 and denominator 0 minus 24

m=20−1210−6

Answer 1

To determine the correct slope (m) of the hot air balloon's ascent, we can use the general formula for slope, which is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

where (x1, y1) and (x2, y2) are two points on the graph.

Let’s analyze the provided options:

1. m = \(\frac{10 - 6}{20 - 12}\)

   • The points are (20, 10) and (12, 6).
   • This calculates to \(m = \frac{4}{8} = \frac{1}{2}\).

2. m = \(\frac{2 - 4}{8 - 4}\)

   • The points are (8, 2) and (4, 4).
   • This calculates to \(m = \frac{-2}{4} = -\frac{1}{2}\).

3. m = \(\frac{12 - 0}{0 - 24}\)

   • The points are (0, 12) and (24, 0).
   • This calculates to \(m = \frac{12}{-24} = -\frac{1}{2}\).

4. m = \(\frac{20 - 12}{10 - 6}\)

   • The points are (10, 20) and (6, 12).
   • This calculates to \(m = \frac{8}{4} = 2\).

To confirm which model correctly reflects the ascent of the balloon, it is essential to understand that a positive slope indicates an upward climb, while a negative slope indicates a decline. Therefore, if the hot air balloon is climbing steadily, the slope must be positive.

Among the options with a positive slope:

• The first option yields a positive slope of \(\frac{1}{2}\).
• The fourth option yields a slope of \(2\).

Thus, both options 1 and 4 reflect a proper ascent. If we had to choose only one, and we assume the first calculation is from the standard interpretation of balloons ascending at a consistent rate, the acceptable answer with a positive slope is m = \(\frac{10 - 6}{20 - 12}\).

Final Answer

m = \(\frac{10 - 6}{20 - 12}\)