Question

answer the questions that follow.
figure 1
figure 2
figure 3
(a) which curve fits the data best?
figure 1 figure 2 figure 3
(b) use the equation of the best - fitting curve from part (a) to predict the amount of the radioactive substance after 80 days. round your answer to the nearest hundredth.
□milligrams

Explanation:

Step1: Analyze the data - point distribution

By observing the scatter - plots, in Figure 1, the curve $y = 200(0.93)^{x}+50$ seems to follow the general trend of the data points well. In Figure 2, the curve $y = 504(0.98)^{x}$ has a steeper decay than the data points suggest. In Figure 3, the linear function $y=-0.26x + 380$ does not capture the non - linear decay pattern of the data. So, Figure 1 has the best - fitting curve.

Step2: Substitute $x = 80$ into the best - fitting equation

The best - fitting equation is $y = 200(0.93)^{x}+50$. Substitute $x = 80$ into it:
$y=200\times(0.93)^{80}+50$.
First, calculate $(0.93)^{80}$. Let $a=(0.93)^{80}$. Using the formula $a = e^{80\ln(0.93)}$.
$\ln(0.93)\approx- 0.0725$, then $80\ln(0.93)\approx80\times(-0.0725)=-5.8$.
$e^{-5.8}\approx0.00304$.
$200\times(0.93)^{80}=200\times0.00304 = 0.608$.
$y=0.608 + 50=50.608\approx50.61$.

Answer:

(a) Figure 1
(b) $50.61$