Math  /  Data & Statistics

QuestionThe table shows the total personal income in the United States (in billions of dollars) for selected years from 1960 and projected to 2024. \begin{tabular}{cc|cc} \hline Year & Income (\ billions) & Year & Income (\$ billions) \\ \hline 1960 & 411.5 & 2008 & 12,100.7 \\ 1970 & 838.8 & 2014 & 14,728.6 \\ 1980 & 2307.9 & 2018 & 19,129.6 \\ 1990 & 4878.6 & 2024 & 22,685.1 \\ 2000 & 8429.7 & & \\ \hline \end{tabular} (a) These data can be modeled by an exponential function. Write the equation of this function, with xasthenumberofyearspast1960and as the number of years past 1960 and yasthetotal as the total y=533.5701.065x Way to gol y=533.570 \cdot 1.065^{x} \quad \text { Way to gol }$ (b) Does the unrounded model overestimate or underestimate the total personal income given in the table for 2018? The model overestimates the projected total. The model underestimates the projected total.
Perfect (c) Graphically determine the year the model predicts total personal income will reach $31\$ 31 trillion. \square Enter a number: Need Help? Read II Watch it

Studdy Solution
Determine when the model predicts income will reach \$31 trillion:
Set y=31,000 y = 31,000 (since 31 trillion is 31,000 billion):
31,000=533.5701.065x 31,000 = 533.570 \cdot 1.065^x
Solve for x x :
1.065x=31,000533.570 1.065^x = \frac{31,000}{533.570}
1.065x58.08 1.065^x \approx 58.08
Take the logarithm of both sides:
xlog(1.065)=log(58.08) x \cdot \log(1.065) = \log(58.08)
xlog(58.08)log(1.065) x \approx \frac{\log(58.08)}{\log(1.065)}
x1.7640.02765.33 x \approx \frac{1.764}{0.027} \approx 65.33
Add this to 1960 to find the year:
1960+65.332025.33 1960 + 65.33 \approx 2025.33
The model predicts income will reach \$31 trillion around the year 2025.
The model underestimates the projected total for 2018, and predicts income will reach \$31 trillion in approximately 2025.

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