Question

the data points show the amount y (in milligrams) of a radioactive substance after decaying for a time x (in days). each figure has the same data points. however, each figure has a different curve fitting the data. the equation for each curve is also shown. answer the questions that follow. figure 1: y - axis from 0 to 500, x - axis from 0 to 200, equation y = 200(0.93)^x + 50, data points plotted. figure 2: y - axis from 0 to 500, x - axis from 0 to 200, equation y = 504(0.98)^x, data points plotted. figure 3: y - axis from 0 to 500, x - axis from 0 to 200, equation y = -0.26x + 380, data points plotted. (a) which curve fits the data best? options: 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:

Part (a)

To determine which curve fits best, we analyze the data points (decaying radioactive substance, so exponential decay is typical). Figure 1’s curve doesn’t align with early points. Figure 3 is linear, but radioactive decay is exponential. Figure 2’s exponential curve ($y = 504(0.98)^x$) follows the data’s downward trend (exponential decay) closely, matching the pattern of a radioactive substance’s decay.

Step1: Identify the best - fitting equation

From part (a), the best - fitting equation is $y = 504(0.98)^x$, where $x$ is the number of days and $y$ is the amount in milligrams.

Step2: Substitute $x = 80$ into the equation

We need to calculate $y$ when $x = 80$. So we substitute $x = 80$ into $y=504(0.98)^{80}$.
First, calculate $(0.98)^{80}$. Using a calculator, $(0.98)^{80}\approx0.18209$.
Then, multiply by 504: $y = 504\times0.18209$.
$y=504\times0.18209 = 504\times(0.1 + 0.08+0.00209)=504\times0.1+504\times0.08 + 504\times0.00209=50.4+40.32+1.05336 = 91.77336\approx91.77$

Answer:

Figure 2

Part (b)