Question

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the table gives the annual global plastic production over time, in years, since 1960
fit an exponential model to this data.
the exponential model indicates that plastic production has per year. the model predicts that plastic production 15 years after 1960 was closest to million metric tons. because the correlation coefficient indicates a correlation between the model and the data, this prediction

Explanation:

Step1: Find the exponential model

The general form of an exponential model is \(y = ab^{x}\), where \(a\) is the initial value and \(b\) is the base. When \(x = 0\), \(y=a\). From the table, when \(x = 0\) (years since 1960), \(y = 5\), so \(a = 5\).
We can use another point \((x,y)=(10,25)\) to find \(b\). Substitute into \(y=ab^{x}\): \(25 = 5b^{10}\). Then \(b^{10}=\frac{25}{5}=5\), and \(b = 5^{\frac{1}{10}}\approx1.1746\). So the model is \(y = 5\times(1.1746)^{x}\). The growth factor \(b\approx1.1746\), which means a growth of about \(17.46\%\) per year.

Step2: Predict for \(x = 15\)

Substitute \(x = 15\) into \(y = 5\times(1.1746)^{15}\). Using the formula \(y=5\times e^{15\ln(1.1746)}\) (since \(a^{x}=e^{x\ln(a)}\)). \(\ln(1.1746)\approx0.16\), \(15\times0.16 = 2.4\), \(e^{2.4}\approx11.023\), \(y=5\times11.023 = 55.115\approx55\)

Step3: Correlation coefficient

Since we are fitting an exponential model to the data points, and if the data points follow an exponential trend (which they seem to as the values are increasing at an increasing rate), the correlation coefficient indicates a strong correlation. And a well - fit exponential model (with a high correlation coefficient) makes the prediction reliable.

Answer:

The exponential model indicates that plastic production has increased by about \(17.5\%\) per year. The model predicts that plastic production 15 years after 1960 was closest to \(55\) million metric tons. Because the correlation coefficient indicates a strong correlation between the model and the data, this prediction is reliable.