To determine whether the relationship between time and the amount of popcorn popped is directly proportional, we need to look for certain characteristics:
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Direct Proportionality Definition: A relationship is directly proportional if it can be expressed in the form \( y = kx \) where \( y \) is the output (amount of popcorn), \( x \) is the input (time), and \( k \) is a constant (rate of popcorn popped per unit of time).
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Origin: For a relationship to be directly proportional, the graph of the relationship must pass through the origin (0,0). This means that when there is zero time elapsed, there should be zero popcorn popped.
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Straight Line: The graph of a directly proportional relationship must be a straight line, where the slope represents the constant rate of change (in this case, the rate at which popcorn is popped over time).
Let's evaluate the responses:
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"Yes, for every 10 minutes there were 4 ounces of popcorn popped.": This suggests a consistent rate of popping. We can derive a constant \( k = \frac{4 \text{ ounces}}{10 \text{ minutes}} = 0.4 \text{ ounces per minute}\). This indicates a directly proportional relationship.
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"No, the graph does not go through the origin.": If this is true, then the relationship cannot be directly proportional.
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"Yes, the input is time and the output is the amount of popcorn popped.": This is simply restating the relationship but does not confirm direct proportionality.
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"No, the graph does not make a straight line.": If the graph were not linear, the relationship would not be directly proportional.
Given the first response indicates a consistent amount of popcorn popped over time, it suggests a direct proportionality. However, if any of the other statements are true (particularly about the graph not passing through the origin or not being a straight line), we would conclude that it is not directly proportional.
Based on the responses provided, if the first statement about the amount of popcorn is accepted, then the answer would be Yes, for every 10 minutes there were 4 ounces of popcorn popped. If additional context indicates that the other statements are true (especially about the graph), then the direct proportionality would be denied.
In conclusion, based on the first response alone, we lean toward it being a direct proportionality unless contradicted by further information.