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What does the 5 represent in Kyle's butterfly collection?

\begin{tabular}{|c|c|c|c|}
\hline[tex]$x$[/tex] & 0 & 2 & 4 \\
\hline[tex]$f(x)$[/tex] & 5 & 20 & 80 \\
\hline
\end{tabular}

A. The number of new butterflies each year
B. The common ratio of butterfly growth
C. The average rate of change of butterfly growth each year
D. The number of butterflies Kyle started with

Sagot :

GinnyAnswer
Let's analyze the table of values given for Kyle's butterfly collection:

[tex]\[ \begin{tabular}{|c|c|c|c|} \hline$x$ & 0 & 2 & 4 \\ \hline$f(x)$ & 5 & 20 & 80 \\ \hline \end{tabular} \][/tex]

In this table, [tex]$x$[/tex] represents time in years, and [tex]$f(x)$[/tex] represents the number of butterflies in Kyle's collection. Let's look at each option to determine what the number 5 represents.

1. The number of new butterflies each year

This option implies that 5 could be the number of new butterflies added each year. However, making close observations, such as [tex]$f(2) = 20$[/tex] and [tex]$f(4) = 80$[/tex], would suggest exponential growth rather than a simple addition of new butterflies each year. This option cannot be correct since the number 5 represents a specific quantity at [tex]$x = 0$[/tex].

2. The common ratio of butterfly growth

This option suggests we are looking for a multiplicative rate between the years. Upon inspection, if we were to look for a common ratio, it must be consistent across the intervals:
- From [tex]$x=0$[/tex] to [tex]$x=2$[/tex], [tex]$f(x)$[/tex] grows from 5 to 20.
- From [tex]$x=2$[/tex] to [tex]$x=4$[/tex], [tex]$f(x)$[/tex] grows from 20 to 80.

The ratio from 5 to 20 is [tex]$ \frac{20}{5} = 4$[/tex], and from 20 to 80 is [tex]$\frac{80}{20} = 4$[/tex]. These calculations confirm a ratio of 4 but do not associate directly with the 5 in context.

3. The average rate of change of butterfly growth each year

This option would calculate the change over the given interval (rate of change). To find the average rate of change:
[tex]\[ \frac{\Delta f(x)}{\Delta x} = \frac{80 - 5}{4 - 0} = \frac{75}{4} = 18.75 \][/tex]
While the average provides information per year, 18.75 is different from 5.

4. The number of butterflies Kyle started with

In the table, at [tex]$x = 0$[/tex], the number of butterflies is 5. This initial amount directly indicates the starting point of the collection.

Given all these considerations:
- The number 5 in the table represents the number of butterflies Kyle started with at year [tex]$x=0$[/tex].

The correct choice is:

The number of butterflies Kyle started with.
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