Gary saw a video on the internet in which dropping mint candies into bottles of soda pop caused the soda pop to spurt immediately from the bottle. He wondered if the height of the spurt was linearly related to the number of mint candies used. He collected data using 1, 3, 5, and 10 mint candies dropped into two-liter bottles of soda pop. The height of the spurt was measured in centimeters. Each quantity of mint candies was tried three times. The data is shown in the table.

Question

Find the mean height of the spurt for 3 mint candies and the mean height of the spurt for 10 candies. Then, use these two data points to estimate the line of fit for the data. What is the BEST interpretation of the slope in context of this problem?
Responses
A The slope is 40, which means that for every mint candy dropped into the bottle of soda pop the minimum height of the spurt increase is 40 cm.The slope is 40, which means that for every mint candy dropped into the bottle of soda pop the minimum height of the spurt increase is 40 cm.
B The slope is 47.1, which means that for every mint candy dropped into the bottle of soda pop the maximum height of spurt increase is 47.1 cm.The slope is 47.1, which means that for every mint candy dropped into the bottle of soda pop the maximum height of spurt increase is 47.1 cm.
C The slope is 43.1, which means that for every mint candy dropped into the bottle of soda pop the height of the spurt increases by 43.1 cm.The slope is 43.1, which means that for every mint candy dropped into the bottle of soda pop the height of the spurt increases by 43.1 cm.
D The slope is 47.1, which means that for every mint candy dropped into the bottle of soda pop the height of the spurt increases by 47.1 cm.

Answer 1

To perform the calculations correctly, I would need the actual data from the experiments conducted by Gary. However, let's go through the process assuming you have the data in the table:

1. Calculate the Mean Heights:

   • For 3 mint candies, you need to sum the heights (from the three trials) and divide by 3.
   • For 10 mint candies, do the same.

2. Estimating the Line of Fit:

   • With the mean heights you calculated for 3 and 10 mint candies, you can find the slope \( m \) using the formula for the slope of a line: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Here, \( (x_1, y_1) \) represents the data point for 3 mint candies (mean height you calculated) and \( (x_2, y_2) \) represents the data point for 10 mint candies (mean height you calculated).

3. Interpreting the Slope:

   • The slope indicates how much the height of the spurt increases with each additional mint candy dropped into the soda.

Now, let's consider the provided options based on the interpretation of the slope:

• A and B mention minimum and maximum heights, which do not align with the given description focusing purely on the height indicated by the slope.
• C and D reference a general increase in the height of the spurt per candy. The differences lie in the slope value.

Ultimately, to choose the BEST interpretation based on the mathematical calculation you perform (including using the correct data for the calculations), you would select the option that accurately reflects the calculated slope. If you find, for instance, that your slope \( m \) turned out to be approximately 43.1, then option C would be your answer. If it was approximately 47.1, then you would select option D.

It's essential to carry out those calculations to determine the precise interpretation.