Question

the data points show the area y (in square kilometers) that a certain forest covers after being inhabited for a time x (in years). 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 figure 2 figure 3 three figures with data points and curves, equations: figure 1: y=2513(0.92)^x, figure 2: y=-75x + 1600, figure 3: y=9.1x² - 275x + 2450 (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 area that the forest covers after it is inhabited for 12 years. round your answer to the nearest hundredth. square kilometers

Explanation:

Part (a)

To determine which curve fits the data best, we visually inspect how closely each curve aligns with the data points. Figure 1's curve (exponential) follows the trend of the data points more closely than Figure 2 (linear) or Figure 3 (quadratic), as the data points seem to follow a smooth exponential decay pattern.

Step1: Identify the equation

The best - fitting curve from part (a) is Figure 1 with the equation \(y = 2513(0.92)^{x}\). We need to find the value of \(y\) when \(x = 12\).

Step2: Substitute \(x = 12\) into the equation

Substitute \(x=12\) into \(y = 2513(0.92)^{x}\). So we have \(y=2513\times(0.92)^{12}\).
First, calculate \((0.92)^{12}\). Using a calculator, \((0.92)^{12}\approx0.367695\).
Then, multiply this by 2513: \(y = 2513\times0.367695\approx2513\times0.3677\) (rounded for easier calculation).
\(2513\times0.3677 = 2513\times(0.3 + 0.06+0.007 + 0.0007)\)
\(=2513\times0.3+2513\times0.06 + 2513\times0.007+2513\times0.0007\)
\(=753.9+150.78+17.591 + 1.7591\)
\(=753.9+150.78 = 904.68\); \(904.68+17.591=922.271\); \(922.271 + 1.7591=924.0301\)
A more accurate calculation using a calculator for \(2513\times(0.92)^{12}\):
\((0.92)^{12}=e^{12\ln(0.92)}\approx e^{12\times(- 0.0833749)}\approx e^{-1.000499}\approx0.3675\)
\(2513\times0.3675 = 2513\times\frac{3675}{10000}=\frac{2513\times3675}{10000}\)
\(2513\times3675=(2500 + 13)\times3675=2500\times3675+13\times3675=9187500+47775 = 9235275\)
\(\frac{9235275}{10000}=923.5275\approx923.53\) (rounded to the nearest hundredth)

Answer:

Figure 1

Part (b)