To create a square root function that models the speed \( s(x) \) as a function of the distance \( x \) from the provided data, we can assume a relationship of the form:
\[ s(x) = k \sqrt{x} \]
where \( k \) is a constant that we will determine using the data from the table.
Let's use the given data points to find the value of \( k \):
- When \( x = 0 \), \( s(0) = 0 \) implies it fits the function.
- When \( x = 1 \), \( s(1) = 6.33 \): \[ 6.33 = k \sqrt{1} \quad \Rightarrow \quad k = 6.33 \]
- When \( x = 2 \), \( s(2) = 8.95 \): \[ 8.95 = k \sqrt{2} \quad \Rightarrow \quad k = \frac{8.95}{\sqrt{2}} \approx 6.34 \]
- When \( x = 3 \), \( s(3) = 10.96 \): \[ 10.96 = k \sqrt{3} \quad \Rightarrow \quad k = \frac{10.96}{\sqrt{3}} \approx 6.34 \]
- When \( x = 4 \), \( s(4) = 12.66 \): \[ 12.66 = k \sqrt{4} \quad \Rightarrow \quad k = \frac{12.66}{2} = 6.33 \]
From the calculations, we find that \( k \) is consistently around 6.33 to 6.34.
Thus, we can conclude that the square root function that models the data can be expressed as:
\[ s(x) \approx 6.33 \sqrt{x} \]
So, the function representation is:
\[ s(x) = 6.33 \sqrt{x} \]
Rounding to the nearest hundredth gives:
\[ s(x) = 6.33 \sqrt{x} \]
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