Question

figure 1
$y = 300(0.99)^x + 130$
figure 2
$y = 546(0.98)^x$
figure 3
$y = 0.03x^2 - 8x + 575$
(a) which curve fits the data best?
\bigcirc figure 1
\bigcirc figure 2
\bigcirc figure 3
(b) use the equation of the best fitting curve from part (a) to predict the amount of the radioactive substance after 50 days. round your answer to the nearest hundredth.
\boxed{\text{milligrams}}

Explanation:

Part (a)

To determine which curve fits the data best, we visually inspect the alignment of the data points with the curve in each figure. Figure 2's curve (with equation \( y = 546(0.98)^x \)) appears to closely follow the trend of the data points, showing a smooth exponential decay - like pattern that matches the distribution of the points better than the curves in Figure 1 (exponential with different parameters) and Figure 3 (quadratic).

Step 1: Identify the equation

From part (a), the best - fitting curve is from Figure 2 with the equation \( y = 546(0.98)^x \), where \( x \) represents the number of days and \( y \) represents the amount of the radioactive substance in milligrams. We need to find the value of \( y \) when \( x = 50 \).

Step 2: Substitute \( x = 50 \) into the equation

Substitute \( x = 50 \) into \( y = 546(0.98)^{50} \). First, calculate \( (0.98)^{50} \). Using a calculator, \( (0.98)^{50}\approx0.364169 \).

Step 3: Calculate \( y \)

Then, multiply this value by 546: \( y = 546\times0.364169\approx546\times0.3642 \) (rounding \( 0.364169 \) to four decimal places for easier calculation). \( 546\times0.3642=(500 + 46)\times0.3642=500\times0.3642+46\times0.3642 = 182.1+16.7532 = 198.8532\approx198.85 \) (rounding to the nearest hundredth).

Answer:

Figure 2

Part (b)