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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?

for every 4 minutes, the balloon rises 200 ft

(1 point)
Responses

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

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

m=20−1210−6
m is equal to the fraction with numerator 20 minus 12 and denominator 10 minus 6

m=12−00−24

5 answers

To find the slope that models the ascent of the hot air balloon, we should use the given information that for every 4 minutes, the balloon rises 200 ft.

The slope \( m \) can be calculated using the formula:

\[ m = \frac{\text{rise}}{\text{run}} \]

In this case, the rise is 200 ft (the vertical change) and the run is 4 minutes (the horizontal change). Therefore,

\[ m = \frac{200}{4} = 50 \]

We will check each of the provided options to see which one, if any, equals 50 when calculated.

  1. Option 1: \( m = \frac{2 - 4}{8 - 4} \) \[ m = \frac{-2}{4} = -0.5 \quad \text{(not correct)} \]

  2. Option 2: \( m = \frac{10 - 6}{20 - 12} \) \[ m = \frac{4}{8} = 0.5 \quad \text{(not correct)} \]

  3. Option 3: \( m = \frac{20 - 12}{10 - 6} \) \[ m = \frac{8}{4} = 2 \quad \text{(not correct)} \]

  4. Option 4: \( m = \frac{12 - 0}{0 - 24} \) \[ m = \frac{12}{-24} = -0.5 \quad \text{(not correct)} \]

None of the provided options yield a slope of 50, which is consistent with the balloon's ascent. Therefore, it appears that none of the examples provided models the slope correctly based on the given ascent rate.

If we must determine which model represents a slope related to the parameters provided, we would note that the correct slope calculation must be verified against the model rather than the options. Since all options produced different incorrect values than expected, they do not successfully represent the balloon's ascent parameters. The correct answer should reflect a slope of 50 (200 ft per 4 minutes).

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

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

m=20−1210−6
m is equal to the fraction with numerator 20 minus 12 and denominator 10 minus 6

m=12−00−24

Let's evaluate the slope \( m \) for each option one by one to see which one is correct.

  1. Option 1: \[ m = \frac{2 - 4}{8 - 4} = \frac{-2}{4} = -0.5 \]

  2. Option 2: \[ m = \frac{10 - 6}{20 - 12} = \frac{4}{8} = 0.5 \]

  3. Option 3: \[ m = \frac{20 - 12}{10 - 6} = \frac{8}{4} = 2 \]

  4. Option 4: \[ m = \frac{12 - 0}{0 - 24} = \frac{12}{-24} = -0.5 \]

Based on the calculations:

  • Option 1 gives \( m = -0.5 \)
  • Option 2 gives \( m = 0.5 \)
  • Option 3 gives \( m = 2 \)
  • Option 4 gives \( m = -0.5 \)

None of the options correctly match the expected slope of 50 (from the original problem of 200 ft every 4 minutes). However, the correct slope calculation based on the problem should equate to 50.

Therefore, while options give various but incorrect results, there is no correct option among them based on the original ascent rate scenario specified. If I need to choose the slope that makes the most logical sense for a positive ascent, Option 3 gives the highest positive slope of 2, but it's still not equivalent to the required rate of 50.

In summary, none of the options provided correctly model the slope for a hot air balloon given it rises 50 ft per minute (200 ft in 4 minutes).

its that divided by 100

Ah, I see! It seems you might be interested in adjusting the values to correspond to a division by 100, possibly interpreting the original rise as a different scale.

Let’s re-evaluate the slopes given the understanding that the values might need to be adjusted as follows:

  1. Option 1: \[ m = \frac{2 - 4}{8 - 4} = \frac{-2}{4} = -0.5 \quad \text{(this remains -0.5)} \]

  2. Option 2: \[ m = \frac{10 - 6}{20 - 12} = \frac{4}{8} = 0.5 \quad \text{(this remains 0.5)} \]

  3. Option 3: \[ m = \frac{20 - 12}{10 - 6} = \frac{8}{4} = 2 \quad \text{(this remains 2)} \]

  4. Option 4: \[ m = \frac{12 - 0}{0 - 24} = \frac{12}{-24} = -0.5 \quad \text{(this remains -0.5)} \]

Since you're suggesting to divide each of these by 100, we can adjust our interpretation accordingly, although the individual calculations remain valid.

To compare them more directly with your original scenario (and assuming you want values divided by 100):

  • Option 1: \( m = -0.5 / 100 \) gives \( -0.005 \)
  • Option 2: \( m = 0.5 / 100 \) gives \( 0.005 \)
  • Option 3: \( m = 2 / 100 \) gives \( 0.02 \)
  • Option 4: \( m = -0.5 / 100 \) gives \( -0.005 \)

So, if you're looking for a model for the increase in feet per minute with respect to a total rise of 200 feet over 4 minutes, that aligns with a rate of:

\[ \frac{200 \text{ ft}}{4 \text{ min}} = 50 \text{ ft/min} \]

When scaled correctly divided by 100, the value would become:

\[ 50 / 100 = 0.5 \]

So let’s revise again: ​when compared directly against the ascent rate of \( 50 ft/min \) compared to 0.5, it perfectly aligns with Option 2, which, when taken relative to time on a scale of 100, becomes:

  • Option 2: \( 0.5 \)

Apologies for any confusion earlier, and thank you for clarifying the intent regarding division by 100! Thus, Option 2 is indeed the closest model to your requirement when interpreted in a different framework.