To determine the best model for the given data, we need to analyze the relationship between the Time (hr) and Weight (mg) variables. The provided results suggest that the log transformation of Weight (mg) as a function of Time (hr) forms a linear relationship. This implies that the original relationship between Time and Weight is exponential.
1. Identify the type of function:
- By converting the weight into a logarithmic scale and noticing that it forms a linear relationship with time, we can conclude that the best model for the data is an exponential function. This means the weight of sodium-24 decreases exponentially over time.
2. Formulate the exponential model:
- The general form of an exponential function is:
[tex]\[ W(t) = a \cdot e^{bt} \][/tex]
where [tex]\( W(t) \)[/tex] is the weight at time [tex]\( t \)[/tex], [tex]\( a \)[/tex] is the initial weight, and [tex]\( b \)[/tex] is the rate of decay.
3. Interpret the constants [tex]\( a \)[/tex] and [tex]\( b \)[/tex]:
- [tex]\( a \)[/tex] is the initial weight: It represents the weight of sodium-24 at time [tex]\( t = 0 \)[/tex].
- [tex]\( b \)[/tex] is the decay constant: It represents the rate at which the sodium-24 decays over time.
From the calculations, we have:
- The transformed weights (logarithms of the original weights): [tex]\[
\begin{aligned}
&4.55912625, \\
&4.30945594, \\
&4.06388535, \\
&3.93573953, \\
&3.7208625, \\
&3.33576958
\end{aligned}
\][/tex]
- The slope ([tex]\( b \)[/tex]) of the line in the log domain: [tex]\[ -0.04893061498389681 \][/tex]
- The intercept ([tex]\( \ln(a) \)[/tex]) of the line in the log domain: [tex]\[ 4.615416084362554 \][/tex]
To find [tex]\( a \)[/tex], we take the exponent of the intercept: [tex]\[
a = e^{4.615416084362554} = 101.02985673177349
\][/tex]
Thus, the best model for the data is:
[tex]\[ W(t) = 101.02985673177349 \cdot e^{-0.04893061498389681 \cdot t} \][/tex]
4. Complete the sentence:
- The best model for the data is an exponential function.
- In the model, [tex]\( a \)[/tex] is the initial weight, and [tex]\( b \)[/tex] is the rate of decay.
Therefore, for the given data, the exponential model effectively describes how the sodium-24 decays over time, with [tex]\( a \)[/tex] representing the initial weight and [tex]\( b \)[/tex] representing the decay constant.
